Derivative of Composite Function

In calculus (a branch of mathematics) the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity.

Composite function:

In mathematics, function composition is the application of one function to the results of another.

(Source: Wikipedia)

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Formula for derivative of composite function:

(dy / dx) = (dy / (du)) * ((du) / (dx))

(f * g) (x) = f' ((g(x)) * g' (x)

Example problems for derivative of composite function

Derivative of composite function example problem 1:

Differentiate the given function y = (x2 - 2)13. Then its equivalent composite function is u = (x2 - 2). Find the first derivative value of y.

Solution:

Given function y = (x2 - 2)13

u = (x2 - 2)

Therfore,

y = (u)13

Formula:

(dy / dx) = ((dy) / (du)) * ((du) / (dx))

Differentiate the function y with respect to u, we get

((dy) / (du)) = 13u12

Differentiate u with respect to x, we get

((du) / (dx)) = 2x

Substitute the above values in the given formula, we get

((dy) / (dx)) = (13u12) (2x)

= 26xu12

Substitute the value of u, we get

((dy) / (dx)) = 26x (x2 - 2)12

Answer:

The final answer is ((dy) / (dx)) = 26x (x2 - 2)12

Derivative of composite function example problem 2:

Differentiate the given function y = (2x - 6)5. Then its equivalent composite function is u = (2x - 6). Find the first derivative value of y.

Solution:

Given function y = (2x - 6)5

u = (2x - 6)

Therfore,

y = (u)5

Formula:

(dy / dx) = ((dy) / (du)) * ((du) / (dx))

Differentiate the function y with respect to u, we get

((dy) / (du)) = 5u4

Differentiate u with respect to x, we get

((du) / (dx)) = 2

Substitute the above values in the given formula, we get

((dy) / (dx)) = (5u4) (2)

= 10u4

Substitute the value of u, we get

((dy) / (dx)) = 10 (2x - 6)4

Answer:

The final answer is ((dy) / (dx)) = 10 (2x - 6)4

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Derivative of composite function example problem 3:

Differentiate the given function y = (2x3 + 10)2. Then its equivalent composite function is u = (2x3 + 10). Find the first derivative value of y.

Solution:

Given function y = (2x3 + 10)2

u = (2x3 + 10)

Therfore,

y = (u)2

Formula:

(dy / dx) = ((dy) / (du)) * ((du) / (dx))

Differentiate the function y with respect to u, we get

((dy) / (du)) = 2u

Differentiate u with respect to x, we get

((du) / (dx)) = 6x2

Substitute the above values in the given formula, we get

((dy) / (dx)) = (2u) (6x2)

= 12x2u

Substitute the value of u, we get

((dy) / (dx)) = 12x2 (2x3 + 10)

Answer:

The final answer is ((dy) / (dx)) = 12x2 (2x3 + 10)

Practice problems for derivative of composite function

Derivative of composite function practice problem 1:

Differentiate the given function y = (14x + 10)3. Then its equivalent composite function is u = (14x + 10). Find the first derivative value of y.

Answer:

The final answer is 42 (14x + 10)2

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Derivative of composite function practice problem 2:

Differentiate the given function y = (x + 2)8. Then its equivalent composite function is u = (x + 2). Find the first derivative value of y.

Answer:

The final answer is 8 (x + 2)7