# Category:Examples of Derivatives of Composite Functions

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This category contains examples of use of **Derivative of Composite Function**.

Let $I, J$ be open real intervals.

Let $g : I \to J$ and $f : J \to \R$ be real functions.

Let $h : I \to \R$ be the real function defined as:

- $\forall x \in \R: \map h x = \map {f \circ g} x = \map f {\map g x}$

Then, for each $x_0 \in I$ such that:

- $g$ is differentiable at $x_0$
- $f$ is differentiable at $\map g {x_0}$

it holds that $h$ is differentiable at $x_0$ and:

- $\map {h'} {x_0} = \map {f'} {\map g {x_0}} \map {g'} {x_0}$

where $h'$ denotes the derivative of $h$.

Using the $D_x$ notation:

- $\map {D_x} {\map f {\map g x} } = \map {D_{\map g x} } {\map f {\map g x} } \map {D_x} {\map g x}$

## Pages in category "Examples of Derivatives of Composite Functions"

The following 23 pages are in this category, out of 23 total.

### D

- Derivative of Composite Function/Examples
- Derivative of Composite Function/Examples/(3x+1)^2
- Derivative of Composite Function/Examples/(3x+1)^4
- Derivative of Composite Function/Examples/(a x^2 + b x + c)^3
- Derivative of Composite Function/Examples/a^sin x
- Derivative of Composite Function/Examples/Exponential of 3 x^2 + 4
- Derivative of Composite Function/Examples/Exponential of a x^2
- Derivative of Composite Function/Examples/Exponential of x^2 + x + 1
- Derivative of Composite Function/Examples/Logarithm of 1 + x^3
- Derivative of Composite Function/Examples/Logarithm of Cosecant of x
- Derivative of Composite Function/Examples/Logarithm of x over x + 1
- Derivative of Composite Function/Examples/Reciprocal of (2x+1)^3
- Derivative of Composite Function/Examples/Reciprocal of Arctangent of x
- Derivative of Composite Function/Examples/Root of 1 + x
- Derivative of Composite Function/Examples/Root of Arcsine of x
- Derivative of Composite Function/Examples/Root of sin x
- Derivative of Composite Function/Examples/Root of x + 1 over x - 1
- Derivative of Composite Function/Examples/Root of x^2 + 1
- Derivative of Composite Function/Examples/Root of x^2 + x + 1
- Derivative of Composite Function/Examples/sin(x^2)
- Derivative of Composite Function/Examples/Sine of 2 x Cubed
- Derivative of Composite Function/Examples/Square of Cosine of a x + b
- Derivative of Sine of x squared