Taking Derivatives _ Calculus _ Khan Academy _ Khan Academy

8/13/2019 Taking Derivatives _ Calculus _ Khan Academy _ Khan Academy

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12/6/13 Taking derivatives | Calculus | Khan Academy | Khan Academy

https://www.khanacademy.org/math/calculus/differential-calculus

C A L C U L U S

C o m m u n i t y Q u e s t i o n s

L i m i t s

T a k i n g d e r i v a t i v e s

D e r i v a t i v e a p p l i c a t i o n s

I n d e f i n i t e a n d d e f i n i t e i n t e g r a l s

S o l i d o f r e v o l u t i o n

S e q u e n c e s , s e r i e s a n d f u n c t i o n

a p p r o x i m a t i o n

A P C a l c u l u s p r a c t i c e q u e s t i o n s

D o u b l e a n d t r i p l e i n t e g r a l s

P a r t i a l d e r i v a t i v e s , g r a d i e n t ,

d i v e r g e n c e , c u r l

L i n e i n t e g r a l s a n d G r e e n ' s

t h e o r e m

S u r f a c e i n t e g r a l s a n d S t o k e s '

t h e o r e m

D i v e r g e n c e t h e o r e m

T a k i n g d e r i v a t i v e s

C a l c u l a t i n g d e r i v a t i v e s . P o w e r r u l e . P r o d u c t a n d q u o t i e n t r u l e s . C h a i n R u l e . I m p l i c i t

d i f f e r e n t i a t i o n . D e r i v a t i v e s o f c o m m o n f u n c t i o n s .

I n t r o d u c t i o n t o d i f f e r e n t i a l

c a l c u l u s

T h e t o p i c t h a t i s n o w k n o w n a s " c a l c u l u s " w a s

r e a l l y c a l l e d " t h e c a l c u l u s o f d i f f e r e n t i a l s " w h e n

f i r s t d e v i s e d b y N e w t o n ( a n d L e i b n i z ) r o u g h l y

f o u r h u n d r e d y e a r s a g o . T o N e w t o n ,

d i f f e r e n t i a l s w e r e i n f i n i t e l y s m a l l " c h a n g e s " i n

n u m b e r s t h a t p r e v i o u s m a t h e m a t i c s d i d n ' t

k n o w w h a t t o d o w i t h . T h i n k t h i s h a s n o

r e l e v e n c e t o y o u ? W e l l h o w w o u l d y o u f i g u r e

o u t h o w f a s t s o m e t h i n g i s g o i n g * r i g h t * a t t h i s

m o m e n t ( y o u ' d h a v e t o f i g u r e o u t t h e v e r y , v e r y

s m a l l c h a n g e i n d i s t a n c e o v e r a n i n f i n i t e l y s m a l l

c h a n g e i n t i m e ) ? T h i s t u t o r i a l g i v e s a g e n t l e

i n t r o d u c t i o n t o t h e w o r l d o f N e w t o n a n d

L e i b n i z .

U s i n g s e c a n t l i n e s l o p e s t o

a p p r o x i m a t e t a n g e n t s l o p e

T h e i d e a o f s l o p e i s f a i r l y s t r a i g h t f o r w a r d - -

( c h a n g e i n v e r t i c a l ) o v e r ( c h a n g e i n h o r i z o n t a l ) .

B u t h o w d o w e m e a s u r e t h i s i f t h e ( c h a n g e i n

h o r i z o n t a l ) i s z e r o ( w h i c h w o u l d b e t h e c a s e

w h e n f i n d i n g t h e s l o p e o f t h e t a n g e n t l i n e . I n

t h i s t u t o r i a l , w e ' l l a p p r o x i m a t e t h i s b y f i n d i n g

t h e s l o p e s o f s e c a n t l i n e s .

I n t r o d u c t i o n t o d e r i v a t i v e s

D i s c o v e r w h a t m a g i c w e c a n d e r i v e w h e n w e

t a k e a d e r i v a t i v e , w h i c h i s t h e s l o p e o f t h e

t a n g e n t l i n e a t a n y p o i n t o n a c u r v e .

S U B S C R I B E

N e w t o n , L e i b n i z , a n d U s a i n B o l t

S l o p e o f a l i n e s e c a n t t o a c u r v e

S l o p e o f a s e c a n t l i n e e x a m p l e 1



A p p r o x i m a t i n g i n s t a n t a n e o u s r a t e o f c h a n g e w o r d p r o b l e m

A p p r o x i m a t i n g e q u a t i o n o f t a n g e n t l i n e w o r d p r o b l e m

S l o p e o f s e c a n t l i n e s

D e r i v a t i v e a s s l o p e o f a t a n g e n t l i n e

T a n g e n t s l o p e a s l i m i t i n g v a l u e o f s e c a n t s l o p e e x a m p l e 1



T a n g e n t s l o p e i s l i m i t i n g v a l u e o f s e c a n t s l o p e

C a l c u l a t i n g s l o p e o f t a n g e n t l i n e u s i n g d e r i v a t i v e d e f i n i t i o n

L E A R N C O A C H

S e a r c h B 4 B O D K H E
https://www.khanacademy.org/profile/_kag5zfmtoYW4tYWNhZGVteXJaCxIIVXNlckRhdGEiTHVzZXJfaWRfa2V5X2h0dHA6Ly9ub3VzZXJpZC5raGFuYWNhZGVteS5vcmcvOWRmODUyMGM4ZDg2MTIzZjIyNWY1Y2U1Njc5MGU5N2EM/https://www.khanacademy.org/https://www.khanacademy.org/coach/reportshttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/calculus--derivatives-2--new-hd-versionhttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/e/tangent-slope-is-limiting-value-of-secant-slopehttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/tangent-slope-as-limiting-value-of-secant-slope-example-3https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/tangent-slope-as-limiting-value-of-secant-slope-example-2https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/tangent-slope-as-limiting-value-of-secant-slope-example-1https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/calculus--derivatives-1--new-hd-versionhttps://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangent/e/slope-of-secant-lineshttps://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangent/v/approximating-equation-of-tangent-line-word-problem-1https://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangent/v/approximating-instantaneous-rate-of-change-word-problem-1https://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangent/v/slope-of-a-secant-line-example-3https://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangent/v/slopes-of-secant-lineshttps://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangent/v/slope-of-a-secant-linehttps://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangent/v/slope-of-a-line-secant-to-a-curvehttps://www.khanacademy.org/math/calculus/differential-calculus/intro_differential_calc/v/newton-leibniz-and-usain-bolthttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_introhttps://www.khanacademy.org/math/calculus/differential-calculus/secant-line-slope-tangenthttps://www.khanacademy.org/math/calculus/differential-calculus/intro_differential_calchttps://www.khanacademy.org/math/calculus/divergence_theorem_topichttps://www.khanacademy.org/math/calculus/multivariable-calculushttps://www.khanacademy.org/math/calculus/line_integrals_topichttps://www.khanacademy.org/math/calculus/partial_derivatives_topichttps://www.khanacademy.org/math/calculus/double_triple_integralshttps://www.khanacademy.org/math/calculus/ap_calc_topichttps://www.khanacademy.org/math/calculus/sequences_series_approx_calchttps://www.khanacademy.org/math/calculus/solid_revolution_topichttps://www.khanacademy.org/math/calculus/integral-calculushttps://www.khanacademy.org/math/calculus/derivative_applicationshttps://www.khanacademy.org/math/calculus/differential-calculushttps://www.khanacademy.org/math/calculus/limits_topichttps://www.khanacademy.org/math/calculus/dhttps://www.khanacademy.org/math/calculus


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V i s u a l i z i n g g r a p h s o f f u n c t i o n s

a n d t h e i r d e r i v a t i v e s

Y o u u n d e r s t a n d t h a t a d e r i v a t i v e c a n b e v i e w e d

a s t h e s l o p e o f t h e t a n g e n t l i n e a t a p o i n t o r t h e

i n s t a n t a n e o u s r a t e o f c h a n g e o f a f u n c t i o n w i t h

r e s p e c t t o x . T h i s t u t o r i a l w i l l d e e p e n y o u r

a b i l i t y t o v i s u a l i z e a n d c o n c e p t u a l i z e

d e r i v a t i v e s t h r o u g h v i d e o s a n d e x e r c i s e s . W e

t h i n k y o u ' l l f i n d t h i s t u t o r i a l i n c r e d i b l y f u n a n d

s a t i s f y i n g ( s e r i o u s l y ) .

P o w e r r u l e

C a l c u l u s i s a b o u t t o s e e m s t r a n g e l y s t r a i g h t

f o r w a r d . Y o u ' v e s p e n t s o m e t i m e u s i n g t h e

d e f i n i t i o n o f a d e r i v a t i v e t o f i n d t h e s l o p e a t a

p o i n t . I n t h i s t u t o r i a l , w e ' l l d e r i v e a n d a p p l y t h e

d e r i v a t i v e f o r a n y t e r m i n a p o l y n o m i a l . B y t h e

e n d o f t h i s t u t o r i a l , y o u ' l l h a v e t h e p o w e r t o

t a k e t h e d e r i v a t i v e o f a n y p o l y n o m i a l l i k e i t ' s

D e r i v a t i v e s 1

T h e d e r i v a t i v e o f f ( x ) = x ^ 2 f o r a n y x

F o r m a l a n d a l t e r n a t e f o r m o f t h e d e r i v a t i v e

F o r m a l a n d a l t e r n a t e f o r m o f t h e d e r i v a t i v e f o r l n x

F o r m a l a n d a l t e r n a t e f o r m o f t h e d e r i v a t i v e e x a m p l e 1

T h e f o r m a l a n d a l t e r n a t e f o r m o f t h e d e r i v a t i v e

I n t e r p r e t i n g s l o p e o f a c u r v e e x e r c i s e

R e c o g n i z i n g s l o p e o f c u r v e s

C a l c u l u s : D e r i v a t i v e s 1

C a l c u l u s : D e r i v a t i v e s 2

D e r i v a t i v e i n t u i t i o n m o d u l e

D e r i v a t i v e i n t u i t i o n

G r a p h s o f f u n c t i o n s a n d t h e i r d e r i v a t i v e s e x a m p l e 1

W h e r e a f u n c t i o n i s n o t d i f f e r e n t i a b l e

I d e n t i f y i n g a f u n c t i o n ' s d e r i v a t i v e e x a m p l e

F i g u r i n g o u t w h i c h f u n c t i o n i s t h e d e r i v a t i v e

G r a p h s o f f u n c t i o n s a n d t h e i r d e r i v a t i v e s

I n t u i t i v e l y d r a w i n g t h e d e r i v a t i v e o f a f u n c t i o n

I n t u i t i v e l y d r a w i n g t h e a n t i d e r i v a t i v e o f a f u n c t i o n

V i s u a l i z i n g d e r i v a t i v e s e x e r c i s e

V i s u a l i z i n g d e r i v a t i v e s

P o w e r r u l e

I s t h e p o w e r r u l e r e a s o n a b l e

D e r i v a t i v e p r o p e r t i e s a n d p o l y n o m i a l d e r i v a t i v e s

P o w e r r u l e
https://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/e/power_rulehttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/e/power_rulehttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/v/derivative-properties-and-polynomial-derivativeshttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/v/is-the-power-rule-reasonablehttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/v/power-rulehttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/e/visualizing_derivativeshttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/visualizing-derivatives-exercisehttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/intuitively-drawing-the-anitderivative-of-a-functionhttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/intuitively-drawing-the-derivative-of-a-functionhttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/e/graphs-of-functions-and-their-derivativeshttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/figuring-out-which-function-is-the-the-derivativehttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/identifying--a-function-s-derivative-examplehttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/where-a-function-is-not-differentiablehttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/graphs-of-functions-and-their-derivatives-example-1https://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/e/derivative_intuitionhttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial/v/derivative-intuition-modulehttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/calculus--derivatives-2https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/calculus--derivatives-1https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/e/recognizing_slopehttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/interpreting-slope-of-a-curve-exercisehttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/e/the-formal-and-alternate-form-of-the-derivativehttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/formal-and-alternate-form-of-the-derivative-example-1https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/formal-and-alternate-form-of-the-derivative-for-ln-xhttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/alternate-form-of-the-derivativehttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/calculus--derivatives-2-5--new-hd-versionhttps://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/e/derivatives_1https://www.khanacademy.org/math/calculus/differential-calculus/derivative_intro/v/calculus--derivatives-2--new-hd-versionhttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorialhttps://www.khanacademy.org/math/calculus/differential-calculus/visualizing-derivatives-tutorial


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s e c o n d n a t u r e !

C h a i n r u l e

Y o u c a n t a k e t h e d e r i v a t i v e s o f f ( x ) a n d g ( x ) , b u t

w h a t a b o u t f ( g ( x ) ) o r g ( f ( x ) ) ? T h e c h a i n r u l e

g i v e s u s t h i s a b i l i t y . B e c a u s e m o s t c o m p l e x a n d

h a i r y f u n c t i o n s c a n b e t h o u g h t o f t h e

c o m p o s i t i o n o f s e v e r a l s i m p l e r o n e s ( o n e s t h a t

y o u c a n f i n d d e r i v a t i v e s o f ) , y o u ' l l b e a b l e t o

t a k e t h e d e r i v a t i v e o f a l m o s t a n y f u n c t i o n a f t e r

t h i s t u t o r i a l . J u s t i m a g i n e .

P r o d u c t a n d q u o t i e n t r u l e s

Y o u c a n f i g u r e o u t t h e d e r i v a t i v e o f f ( x ) . Y o u ' r e

a l s o g o o d f o r g ( x ) . B u t w h a t a b o u t f ( x ) t i m e s

g ( x ) ? T h i s i s w h a t t h e p r o d u c t r u l e i s a l l a b o u t .

T h i s t u t o r i a l i s a l l a b o u t t h e p r o d u c t r u l e . I t a l s o

c o v e r s t h e q u o t i e n t r u l e ( w h i c h r e a l l y i s j u s t a

s p e c i a l c a s e o f t h e p r o d u c t r u l e ) .

P r o o f : d / d x ( x ^ n )

P r o o f : d / d x ( s q r t ( x ) )

P o w e r r u l e i n t r o d u c t i o n

D e r i v a t i v e s o f s i n x , c o s x , t a n x , e ^ x a n d l n x

S p e c i a l d e r i v a t i v e s

C h a i n r u l e i n t r o d u c t i o n

C h a i n r u l e d e f i n i t i o n a n d e x a m p l e

C h a i n r u l e w i t h t r i p l e c o m p o s i t i o n

C h a i n r u l e f o r d e r i v a t i v e o f 2 ^ x

D e r i v a t i v e o f l o g w i t h a r b i t r a r y b a s e

C h a i n r u l e e x a m p l e u s i n g v i s u a l f u n c t i o n d e f i n i t i o n s

C h a i n r u l e e x a m p l e u s i n g v i s u a l i n f o r m a t i o n

C h a i n r u l e o n t w o f u n c t i o n s

E x t r e m e d e r i v a t i v e w o r d p r o b l e m ( a d v a n c e d )

T h e c h a i n r u l e

C h a i n r u l e e x a m p l e s

E v e n m o r e c h a i n r u l e

M o r e e x a m p l e s u s i n g m u l t i p l e r u l e s

D e r i v a t i v e s o f s i n x , c o s x , t a n x , e ^ x a n d l n x

S p e c i a l d e r i v a t i v e s

A p p l y i n g t h e p r o d u c t r u l e f o r d e r i v a t i v e s

P r o d u c t r u l e f o r m o r e t h a n t w o f u n c t i o n s

P r o d u c t r u l e

Q u o t i e n t r u l e f r o m p r o d u c t r u l e
https://www.khanacademy.org/math/calculus/differential-calculus/product_rule/v/quotient-rule-from-product-rulehttps://www.khanacademy.org/math/calculus/differential-calculus/product_rule/v/quotient-rule-from-product-rulehttps://www.khanacademy.org/math/calculus/differential-calculus/product_rule/e/product_rulehttps://www.khanacademy.org/math/calculus/differential-calculus/product_rule/v/product-rule-for-more-than-two-functionshttps://www.khanacademy.org/math/calculus/differential-calculus/product_rule/v/applying-the-product-rule-for-derivativeshttps://www.khanacademy.org/math/calculus/differential-calculus/product_rule/e/special_derivativeshttps://www.khanacademy.org/math/calculus/differential-calculus/product_rule/v/derivatives-of-sin-x--cos-x--tan-x--e-x-and-ln-xhttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/derivatives--part-9https://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/even-more-chain-rulehttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/chain-rule-exampleshttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/the-chain-rulehttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/extreme-derivative-word-problem--advancedhttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/e/chain_rule_1https://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/chain-rule-on-two-functions-example-2https://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/chain-rule-on-two-functions-example-1https://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/derivative-of-log-with-arbitrary-basehttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/chain-rule-for-derivative-of-2-xhttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/chain-rule-with-triple-compositionhttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/chain-rule-definition-and-examplehttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/chain-rule-introductionhttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/e/special_derivativeshttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule/v/derivatives-of-sin-x--cos-x--tan-x--e-x-and-ln-xhttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/v/calculus--derivatives-3https://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/v/proof--d-dx-sqrt-xhttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/v/proof--d-dx-x-nhttps://www.khanacademy.org/math/calculus/differential-calculus/power_rule_tutorial/e/power_rulehttps://www.khanacademy.org/math/calculus/differential-calculus/product_rulehttps://www.khanacademy.org/math/calculus/differential-calculus/chain_rule


4/4



I m p l i c i t d i f f e r e n t i a t i o n

L i k e p e o p l e , m a t h e m a t i c a l r e l a t i o n s a r e n o t

a l w a y s e x p l i c i t a b o u t t h e i r i n t e n t i o n s . I n t h i s

t u t o r i a l , w e ' l l b e a b l e t o t a k e t h e d e r i v a t i v e o f

o n e v a r i a b l e w i t h r e s p e c t t o a n o t h e r e v e n w h e n

t h e y a r e i m p l i c i t l y d e f i n e d ( l i k e " x ^ 2 + y ^ 2 = 1 " ) .

P r o o f s o f d e r i v a t i v e s o f c o m m o n

f u n c t i o n s

W e t o l d y o u a b o u t t h e d e r i v a t i v e s o f m a n y

f u n c t i o n s , b u t y o u m i g h t w a n t p r o o f t h a t w h a t

w e t o l d y o u i s a c t u a l l y t r u e . T h a t ' s w h a t t h i s

t u t o r i a l t r i e s t o d o !

Q u o t i e n t r u l e f o r d e r i v a t i v e o f t a n x

Q u o t i e n t r u l e

U s i n g t h e p r o d u c t r u l e a n d t h e c h a i n r u l e

P r o d u c t r u l e

Q u o t i e n t r u l e a n d c o m m o n d e r i v a t i v e s

E q u a t i o n o f a t a n g e n t l i n e


S h o w i n g e x p l i c i t a n d i m p l i c i t d i f f e r e n t i a t i o n g i v e s a m e r e s u l t

I m p l i c i t d e r i v a t i v e o f ( x - y ) ^ 2 = x + y - 1

I m p l i c i t d e r i v a t i v e o f y = c o s ( 5 x - 3 y )

I m p l i c i t d e r i v a t i v e o f ( x ^ 2 + y ^ 2 ) ^ 3 = 5 x ^ 2 y ^ 2

F i n d i n g s l o p e o f t a n g e n t l i n e w i t h i m p l i c i t d i f f e r e n t i a t i o n

I m p l i c i t d e r i v a t i v e o f e ^ ( x y ^ 2 ) = x - y

D e r i v a t i v e o f x ^ ( x ^ x )


P r o o f : d / d x ( l n x ) = 1 / x

P r o o f : d / d x ( e ^ x ) = e ^ x

P r o o f s o f d e r i v a t i v e s o f l n ( x ) a n d e ^ x

A B O U T

O u r M i s s i o n

B l o g

O u r T e a m

O u r I n t e r n s

O u r C o n t e n t S p e c i a l i s t s

O u r S u p p o r t e r s

D i s c o v e r y L a b

O u r M C A T C o n t e n t

C r e a t o r s

H E L P

R e p o r t a P r o b l e m

F A Q

C O N T A C T U S

C o n t a c t

P r e s s

C O A C H I N G

C o a c h R e p o r t s

C o a c h R e s o u r c e s

C a s e S t u d i e s

C o m m o n C o r e

C A R E E R S

F u l l T i m e

I n t e r n s h i p s

M O R E

D o n a t e

C o n t r i b u t e

S h o p K A

I N T E R N A T I O N A L

C h a n g e l a n g u a g e

T r a n s l a t e o u r c o n t e n t

S O C I A L

F a c e b o o k

T w i t t e r

L i f e a t K A

T e r m s o f U s e P r i v a c y P o l i c y 2 0 1 3 K h a n A c a d e m y

E x c e p t w h e r e n o t e d , a l l r i g h t s r e s e r v e d
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