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Higher Order Derivatives
Implicit Differentiation
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 1 / 19
For today
1 Higher Order Derivatives
2 Implicit Differentiation
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 2 / 19
Higher Order Derivatives
If f is differentiable, then the derivative of f is called the second derivative of fand is denoted f .
We can continue to obtain the third derivative, f , the fourth derivative, f (4),and other higher derivatives.
DefinitionThe n-th derivative of the function f , denoted f (n), is the derivative of the(n 1)-th derivative of f , that is,
f (n)(x) = limx0
f (n1)(x+ x) f (n1)(x)x
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 4 / 19
Higher Order Derivatives
Remarks
1 The n in f (n) is called the order of the derivative.
2 The derivative of a function f is also called the first derivative of f .
3 The function f is sometimes written as f (0)(x).
4 Other notations: Dxn[ f (x)] ,dnydxn
,dn
dxn[ f (x)] , y(n)
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 5 / 19
Higher Order Derivatives
Example
Find f (n)(x) for all n N where f (x) = x6 x4 3x3 + 2x2 4.
Solution. We differentiate repeatedly and obtain
f (x) = 6x5 4x3 9x2 + 4x,f (x) = 30x4 12x2 18x+ 4,f (x) = 120x3 24x 18,f (4)(x) = 360x2 24,f (5)(x) = 720x,
f (6)(x) = 720,
f (n)(x) = 0 for all n 7.
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 6 / 19
Higher Order Derivatives
Example
Find f (4)(x) if f (x) =2x 3.
Solution.f (x) = 1
2(2x 3) 12 (2) = (2x 3) 12
f (x) = 12(2x 3) 32 (2) = (2x 3) 32
f (x) = 32(2x 3) 52 (2) = 3 (2x 3) 52
f (4)(x) = 152(2x 3) 72 (2) = 15 (2x 3) 72
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 7 / 19
Equations in two variables are used to define a function explicitly or implicitly.
y = x2 1 defines f (x) = x2 1 (explicitly).
y2 = x+ 1 defines two functions of x: (implicitly)
f1(x) =x+ 1
f2(x) = x+ 1.
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 9 / 19
Consider: If x3 2xy 3y 6 = 0, how do we find dydx
?
SolutionFrom x3 2xy 3y 6 = 0, we can define y explicitly in terms of x as
y =x3 62x+ 3
.
Thus,dydx
=(2x+ 3)(3x2) (x3 6)(2)
(2x+ 3)2=
4x3 + 9x2 + 12(2x+ 3)2
.
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 10 / 19
Implicit Differentiation
Question: Suppose we have x4y3 7xy = 7 or tan(x2 2xy) = y.How do we find
dydx
?
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 11 / 19
Implicit Differentiation
To finddydx
using implicit differentiation, we
1 think of the variable y as a differentiable function of the variable x,
2 differentiate both sides of the equation, using the chain rule where necessary,
and
3 solve fordydx
.
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 12 / 19
Implicit Differentiation
Example
Finddydx
if x3 2xy 3y 6 = 0.
Solution.
Dx[x3 2xy 3y 6] = Dx[0]Dx[x3] Dx[2xy] Dx[3y] Dx[6] = Dx[0]3x2 2
(1 y+ x dy
dx
) 3 dy
dx 0 = 0
3x2 2y 2x dydx 3 dy
dx= 0
2x dydx
+ 3 dydx
= 3x2 2ydydx
(2x+ 3) = 3x2 2y
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 13 / 19
Implicit Differentiation
dydx
=3x2 2y2x+ 3
=3x2 2 x362x+3
2x+ 3
=3x2 (2x+ 3) 2(x3 6)
(2x+ 3)2
=4x3 + 9x2 + 12(2x+ 3)2
.
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 14 / 19
Implicit Differentiation
Example
Finddydx
if x4y3 7xy = 7.
Solution.
Dx[x4y3 7xy] = Dx[7]4x3y3 + 3x4y2 dy
dx 7(y+ x dy
dx
)= 0
dydx
=4x3y3 + 7y3x4y2 7x .
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 15 / 19
Implicit Differentiation
Example
Finddydx
if tan(x2 2xy) = y.
Solution.
Dx[tan(x2 2xy)] = Dx[y]sec2(x2 2xy)
[2x 2
(y+ x dy
dx
)]=
dydx
Thus,
dydx
=2x sec2(x2 2xy) 2y sec2(x2 2xy)
1+ 2x sec2(x2 2xy) .
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 16 / 19
Higher Order Derivatives
Example
Determined2ydx2
if xy2 = y 2.
Solution.
Dx[xy2] = Dx[y 2]y2 + x 2y dy
dx=
dydx
dydx
=y2
1 2xy
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 17 / 19
We havedydx
=y2
1 2xy .
= d2ydx2
=(1 2xy)(2y dydx ) (y2)[2(y+ x dydx )]
(1 2xy)2
=(1 2xy)
[2y y212xy
] (y2)
[2(y+ x y212xy
)](1 2xy)2 .
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 18 / 19
Exercises
1 Determine D3x [ cos(4x) ].
2 Finddydx
if
sin(x+ y) = xy2 2x3.3 Find the equation of the tangent line to the graph of
3x cos y = 1+ x2 +2y+ 1 at P(1, 0).
4 Find the fourth derivative ofx6 1x2
.
Institute of Mathematics (UP Diliman) Higher Order, Implicit Differentiation Mathematics 53 19 / 19
Higher Order DerivativesImplicit Differentiation