# Derivative of Composite Function/Examples

## Examples of Derivatives of Composite Functions

### Example: $\paren {3 x + 1}^2$

$\map {\dfrac \d {\d x} } {\paren {3 x + 1}^2} = 6 \paren {3 x + 1}$

### Example: $\map \sin {x^2}$

$\map {\dfrac \d {\d x} } {\map \sin {x^2} } = 2 x \map \cos {x^2}$

### Example: $\sqrt {1 + x}$

$\map {\dfrac \d {\d x} } {\sqrt {1 + x} } = \dfrac 1 {2 \sqrt {1 + x} }$

### Example: $\sqrt {\sin x}$

$\map {\dfrac \d {\d x} } {\sqrt {\sin x} } = \dfrac {\cos x} {2 \sqrt {\sin x} }$

### Example: $\paren {3 x + 1}^4$

$\map {\dfrac \d {\d x} } {\paren {3 x + 1}^4} = 12 \paren {3 x + 1}^3$

### Example: $\dfrac 1 {\paren {2 x + 1}^3}$

$\map {\dfrac \d {\d x} } {\dfrac 1 {\paren {2 x + 1}^3} } = -\dfrac 6 {\paren {2 x + 1}^4}$

### Example: $\sqrt {x^2 + x + 1}$

$\map {\dfrac \d {\d x} } {\sqrt {x^2 + x + 1} } = \dfrac {2 x + 1} {2 \sqrt {x^2 + x + 1} }$

### Example: $\sin^3 2 x$

$\map {\dfrac \d {\d x} } {\sin^3 2 x} = 6 \sin^2 2 x \cos 2 x$

### Example: $\map {\cos^2} {a x + b}$

$\map {\dfrac \d {\d x} } {\map {\cos^2} {a x + b} } = -2 a \map \cos {a x + b} \map \sin {a x + b}$

### Example: $\sqrt {\arcsin x}$

$\map {\dfrac \d {\d x} } {\sqrt {\arcsin x} } = \dfrac 1 {2 \sqrt {\paren {1 - x^2} \arcsin x } }$

### Example: $\dfrac 1 {\arctan x}$

$\map {\dfrac \d {\d x} } {\dfrac 1 {\arctan x} } = -\dfrac 1 {\paren {1 + x^2} \arctan^2 x}$

### Example: $e^{a x^2}$

$\map {\dfrac \d {\d x} } {e^{a x^2} } = 2 a x e^{a x^2}$

### Example: $\map \ln {1 + x^3}$

$\map {\dfrac \d {\d x} } {\map \ln {1 + x^3} } = \dfrac {3 x^2} {1 + x^3}$

### Example: $\ln \cosec x$

$\map {\dfrac \d {\d x} } {\ln \cosec x} = -\cot x$

### Example: $\sqrt {\dfrac {x + 1} {x - 1} }$

$\map {\dfrac \d {\d x} } {\sqrt {\dfrac {x + 1} {x - 1} } } = -\dfrac 1 {\sqrt {\paren {x + 1} \paren {x - 1}^3} }$

### Example: $\ln \dfrac x {x + 1}$

$\map {\dfrac \d {\d x} } {\ln \dfrac x {x + 1} } = \dfrac 1 {x \paren {x + 1} }$

### Example: $e^{3 x^2 + 4}$

$\map {\dfrac \d {\d x} } {e^{3 x^2 + 4} } = 6 x e^{3 x^2 + 4}$

### Example: $e^{x^2 + x + 1}$

$\map {\dfrac \d {\d x} } {e^{x^2 + x + 1} } = \paren {2 x + 1} e^{x^2 + x + 1}$

### Example: $\sqrt {x^2 + 1}$

$\map {\dfrac \d {\d x} } {\sqrt {x^2 + 1} } = \dfrac x {\sqrt {x^2 + 1} }$

### Example: $\paren {a x^2 + b x + c}^3$

$\map {\dfrac \d {\d x} } {\paren {a x^2 + b x + c}^3} = 3 \paren {2 a x + b} \paren {a x^2 + b x + c}^2$

### Example: $a^{\sin x}$

$\map {\dfrac \d {\d x} } {a^{\sin x} } = \cos x a^{\sin x} \ln a$