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11 . DIFFERENTI AT ION

Synopsis :1. Let f be a function defined in a neighbourhood of a real number a. Then f is said to be differentiable or

derivable at a if ax )a(f)x(fLtax exists. The limit is called the derivative or differential coefficient of f at a. It is

denoted by f I(a).

2. If f is differentiable at a, then f I(a) =h

)a(f)ha(fLt

0h

+

.

3. Let f be a function defined on a neighbourhood of a real number a. Then f is said to be right differentiable at

a ifax

)a(f)x(fLt

ax

+exists. The limit is called the right derivative of f at a. It is denoted by f I(a + ).

4. Similarly the left derivative of a function f at a is defined as f I(a ) =ax

)a(f)x(fLt

ax

.

5. Let f be a function defined on [a, b]. Then f is said to be differentiable on [a, b] ifi) f is differentiable at c where c (a, b)ii) f is right differentiable at aiii) f is left differentiable at b.

6. If a function f is differentiable at a, then f is continuous at a.

7. If c is a constant then }c{dxd = 0.

8. 1}x{dxd = .

9. 1nn nx}x{dxd = .

10.x2

1}x{

dxd = .

f(x) f I(x)

c R 0

x 1

xn; n N nx n1

xn; n R nx n1

ex ex

ax, a R + axloga

logx 1/x

|x| |x|/x, x 0log|x| 1/x

xx xx(1 + logx)

sinx cosx

cosx sinx

tanx sec 2x

cotx cosec 2x

secx secxtanx

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cosecx cosecxcotx

Sin 1x 2x1

1

Cos 1 x 2x1

1

Tan 1 x 2x1

1+

Cot 1 x 2x1

1+

Sec 1 x 1x|x|

12

Cosec 1 x 1x|x|

12

Sinhx coshxCoshx sinhxTanhx sech 2xCothx cosech 2x Sechx sechxtanhx Cosechx cosechxcothx Sinh 1 x

2

x1

1

+

Cosh 1 x 1x

12

Tanh 1 x 2x1

1

(|x|1)

Sech 1 x 2x1|x|

1

Cosech1

x 1x|x|

12 +

11. { }

+= )x(flog)x(g

)x(f)x(f

)x(g)x(f)x(fdxd I

I)x(g)x(g

12. { } [ ] )x(f)x(flog1)x(f)x(fdxd I)x(f)x(f +=

13. If y = f(x) y, then]ylog1)[x(f

)x(fy)]x(flogy1)[x(f

)x(fydxdy I2I2

=

=

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14. If y = y)x(f + , then1y2)x(f

dxdy I

= .

15. If y = f(x) +y

1, then

1y

)x(fy

dx

dy2

I2

+

=

16. If f(x, y) = c is an implicit function, thenyf

/ xf

dxdy

= .

17. If f(x + y) = f(x)f(y), x,y R and f(x) 0, f(a) = k, f I(0) exists, then f I(a) = kf I(0).

18. If y = )....x(f)x(f)x(f , then )x(fdxdy I= .

19. If x myn = (x + y) m + n , thenxy

dxdy = .

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