Learn Online the Chain Rule

Rule for solving the derivative of a composition of two functions.

If y is a function of   u   and   u is a function of   x, then y is a function of x.

The chain rule notify us how to find the derivative of  y   with respect to x

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Through online to help you learn better and faster. In fact they not only help you with accepting Math concepts better but also help you with your Math homework and assignments.

Chain rule formula:

f (x) = g (h(x))

f’(x) = g’ (h(x) h’(x)

(dy) / (dx) = (dy)/(du) xx (du)/(dx)

Learn online the Chain Rule - Examples:

Learn online the Chain Rule - Example 1:

Differentiate y = (3x + 1)2

D (3x + 1)2 = 2(3x +1) 2-1 D (3x + 1)

= 2 (3x + 1) (3)

= 6 (3x + 1).

Learn online the Chain Rule - Example 2:

Differentiate y = (1 -4x + 7x5) 30

The outer is the 30th power and the inner is (1 – 4x +7x5). Differentiate the 30th power first, leaving (1 – 4x +7x5) unchanged. Then differentiate (1 – 4x +7x5). Thus,

D ( 1– 4x + 7x5) = 30 ( 1-4x+7x5) 30-1 D( 1-4x+7x5)

= 30 (1-4x+7x5) 20-1 (-4+35x4)

= 30(35x4 – 4)(1-4x+7x5)29.

= (4x+x-5)1/2

Learn online the Chain Rule - Example 3:

Differentiate the following:

Y = sin(x2+3)

Let u = x2 +3 so y = sin u

(du)/(dx) =2x         (dy)/(du) = cos u

(dy)/(dx)  = cos ux 2x

(dy)/(dx) = 2xcos(x2+3)

Example 4:

Y = ln (3x3-4x+2)

U = 3x3 - 4x + 2         y = ln u

(du)/(dx) = 9x2 -4          (dy)/(du) = 1/u

(dy)/(dx)  = 1/u xx (9x2 – 4)

(dy)/(dx) = (9x^2 - 4)/(3x^3 - 4x + 2)

Note that (dy)/ (dx) = (u')/u

In general if

Y = ln f(x) then (dy)/(dx) = (f'(x))/f(x)

This is a useful result to remember

Example 5:

Differentiate the following:

Y = sin 2 4x

Now sin 2 4x = (sin 4x)2

Let u = sin 4x                  y = u2

(du)/(dx) = 4 cos 4x       (dy)/(du) =2u

(dy)/(dx) = 2u x 4cos 4x

(dy)/(dx)  = 8 sin 4x cos 4x

Learn online the Chain Rule - Practice Problems:

Practice problem 1:

Differentiate y = sqrt (3e^x + 4)

Answer:

(3e^x)/ (2(sqrt(3e^x + 4)))

Practice problem 2

Differentiate  y = (3x2+1)2

Answer:

36x3 + 12x