Calculus Examples

$y = {\sin (5 x)}^{8 x}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${\sin (5 x)}^{8 x}$ as $e^{\ln ({\sin (5 x)}^{8 x})}$ .

$\frac{d}{d x} [e^{\ln ({\sin (5 x)}^{8 x})}]$

Step 1.2

Expand $\ln ({\sin (5 x)}^{8 x})$ by moving $8 x$ outside the logarithm.

$\frac{d}{d x} [e^{8 x \ln (\sin (5 x))}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 8 x \ln (\sin (5 x))$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $8 x \ln (\sin (5 x))$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [8 x \ln (\sin (5 x))]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [8 x \ln (\sin (5 x))]$

Step 2.3

Replace all occurrences of $u_{1}$ with $8 x \ln (\sin (5 x))$ .

$e^{8 x \ln (\sin (5 x))} \frac{d}{d x} [8 x \ln (\sin (5 x))]$

Step 3

Differentiate using the Constant Multiple Rule.

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x \ln (\sin (5 x))$ with respect to $x$ is $8 \frac{d}{d x} [x \ln (\sin (5 x))]$ .

$e^{8 x \ln (\sin (5 x))} (8 \frac{d}{d x} [x \ln (\sin (5 x))])$

Step 3.2

Move $8$ to the left of $e^{8 x \ln (\sin (5 x))}$ .

$8 \cdot e^{8 x \ln (\sin (5 x))} \frac{d}{d x} [x \ln (\sin (5 x))]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (\sin (5 x))$ .

$8 e^{8 x \ln (\sin (5 x))} (x \frac{d}{d x} [\ln (\sin (5 x))] + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \sin (5 x)$ .

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Step 5.1

To apply the Chain Rule, set $u_{2}$ as $\sin (5 x)$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\sin (5 x)]) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 5.2

The derivative of $\ln (u_{2})$ with respect to $u_{2}$ is $\frac{1}{u_{2}}$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\frac{1}{u_{2}} \frac{d}{d x} [\sin (5 x)]) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 5.3

Replace all occurrences of $u_{2}$ with $\sin (5 x)$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\frac{1}{\sin (5 x)} \frac{d}{d x} [\sin (5 x)]) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 6

Convert from $\frac{1}{\sin (5 x)}$ to $\csc (5 x)$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\csc (5 x) \frac{d}{d x} [\sin (5 x)]) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 5 x$ .

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Step 7.1

To apply the Chain Rule, set $u_{3}$ as $5 x$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\csc (5 x) (\frac{d}{d u_{3}} [\sin (u_{3})] \frac{d}{d x} [5 x])) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 7.2

The derivative of $\sin (u_{3})$ with respect to $u_{3}$ is $\cos (u_{3})$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\csc (5 x) (\cos (u_{3}) \frac{d}{d x} [5 x])) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 7.3

Replace all occurrences of $u_{3}$ with $5 x$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\csc (5 x) (\cos (5 x) \frac{d}{d x} [5 x])) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 8

Differentiate.

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Step 8.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\csc (5 x) \cos (5 x) (5 \frac{d}{d x} [x])) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 8.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\csc (5 x) \cos (5 x) (5 \cdot 1)) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $5$ by $1$ .

$8 e^{8 x \ln (\sin (5 x))} (x (\csc (5 x) \cos (5 x) \cdot 5) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 8.3.2

Move $5$ to the left of $\csc (5 x) \cos (5 x)$ .

$8 e^{8 x \ln (\sin (5 x))} (x (5 \cdot (\csc (5 x) \cos (5 x))) + \ln (\sin (5 x)) \frac{d}{d x} [x])$

Step 8.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$8 e^{8 x \ln (\sin (5 x))} (x (5 \csc (5 x) \cos (5 x)) + \ln (\sin (5 x)) \cdot 1)$

Step 8.5

Multiply $\ln (\sin (5 x))$ by $1$ .

$8 e^{8 x \ln (\sin (5 x))} (x (5 \csc (5 x) \cos (5 x)) + \ln (\sin (5 x)))$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$8 e^{8 x \ln (\sin (5 x))} (x (5 \csc (5 x) \cos (5 x))) + 8 e^{8 x \ln (\sin (5 x))} \ln (\sin (5 x))$

Step 9.2

Multiply $5$ by $8$ .

$40 e^{8 x \ln (\sin (5 x))} x \csc (5 x) \cos (5 x) + 8 e^{8 x \ln (\sin (5 x))} \ln (\sin (5 x))$

Step 9.3

Reorder terms.

$40 e^{8 x \ln (\sin (5 x))} x \cos (5 x) \csc (5 x) + 8 e^{8 x \ln (\sin (5 x))} \ln (\sin (5 x))$