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Derivative of composite functions

Derivative of Composite Functions A composite function , $f(g(x))$, is a function defined by the composition of two functions: $f(x)$ and $g(x)$. To find...

Derivative of Composite Functions#

A composite function, $f(g(x))$ , is a function defined by the composition of two functions: $f(x)$ and $g(x)$ .

To find the derivative of $f(g(x))$ , we need to consider the derivative of $f(x)$ with respect to $x$ and the derivative of $g(x)$ with respect to $x$ .

The derivative of $f(x)$ is denoted by $f'(x)$ , and the derivative of $g(x)$ is denoted by $g'(x)$ .

The derivative of $f(g(x))$ is calculated by applying the chain rule:

$(f(g(x)))' = f'(g(x)) \cdot g'(x)$

Example:

Let $f(x) = (x^2 + 1)^3$ and $g(x) = x + 2$ . Then:

$f(g(x)) = (x + 2)^2 + 1$

The derivative of $f(g(x))$ with respect to $x$ is:

$f'(g(x)) = (x + 2)(2x) + 0 = 2x^2 + 2x$

Therefore, $f'(x) = (x^2 + 2x)(x + 2)$

Derivative of Composite Functions#

A composite function,

f(g(x))

, is a function defined by the composition of two functions:

f(x)

and

g(x)

To find the derivative of

f(g(x))

, we need to consider the derivative of

f(x)

with respect to

x

and the derivative of

g(x)

with respect to

x

The derivative of

f(x)

is denoted by

f'(x)

, and the derivative of

g(x)

is denoted by

g'(x)

The derivative of $f(g(x))$ is calculated by applying the chain rule:

(f(g(x)))' = f'(g(x)) \cdot g'(x)

Example:

Let

f(x) = (x^2 + 1)^3

and

g(x) = x + 2

. Then:

f(g(x)) = (x + 2)^2 + 1

The derivative of

f(g(x))

with respect to

x

is:

f'(g(x)) = (x + 2)(2x) + 0 = 2x^2 + 2x

Therefore,

f'(x) = (x^2 + 2x)(x + 2)

Derivative of composite functions

Derivative of Composite Functions#

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Derivative of Composite Functions#