Calculus Examples

${\cos (3 x)}^{\sin (3 x)}$

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${\cos (3 x)}^{\sin (3 x)}$ as $e^{\ln ({\cos (3 x)}^{\sin (3 x)})}$ .

$\frac{d}{d x} [e^{\ln ({\cos (3 x)}^{\sin (3 x)})}]$

Step 1.2

Expand $\ln ({\cos (3 x)}^{\sin (3 x)})$ by moving $\sin (3 x)$ outside the logarithm.

$\frac{d}{d x} [e^{\sin (3 x) \ln (\cos (3 x))}]$

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 $g (x) = \sin (3 x) \ln (\cos (3 x))$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (3 x) \ln (\cos (3 x))$ .

$\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [\sin (3 x) \ln (\cos (3 x))]$

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 $a$ = $e$ .

$e^{u_{1}} \frac{d}{d x} [\sin (3 x) \ln (\cos (3 x))]$

Step 2.3

Replace all occurrences of $u_{1}$ with $\sin (3 x) \ln (\cos (3 x))$ .

$e^{\sin (3 x) \ln (\cos (3 x))} \frac{d}{d x} [\sin (3 x) \ln (\cos (3 x))]$

Step 3

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 $f (x) = \sin (3 x)$ and $g (x) = \ln (\cos (3 x))$ .

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

Step 4

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 $g (x) = \cos (3 x)$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $\cos (3 x)$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sin (3 x) (\frac{d}{d u_{2}} [\ln (u_{2})] \frac{d}{d x} [\cos (3 x)]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 4.2

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

$e^{\sin (3 x) \ln (\cos (3 x))} (\sin (3 x) (\frac{1}{u_{2}} \frac{d}{d x} [\cos (3 x)]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 4.3

Replace all occurrences of $u_{2}$ with $\cos (3 x)$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sin (3 x) (\frac{1}{\cos (3 x)} \frac{d}{d x} [\cos (3 x)]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 5

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sin (3 x) (\sec (3 x) \frac{d}{d x} [\cos (3 x)]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 3 x$ .

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Step 6.1

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

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

Step 6.2

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

$e^{\sin (3 x) \ln (\cos (3 x))} (\sin (3 x) \sec (3 x) (- \sin (u_{3}) \frac{d}{d x} [3 x]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 6.3

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

$e^{\sin (3 x) \ln (\cos (3 x))} (\sin (3 x) \sec (3 x) (- \sin (3 x) \frac{d}{d x} [3 x]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 7

Raise $\sin (3 x)$ to the power of $1$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- (\sin^{1} (3 x) \sin (3 x)) \frac{d}{d x} [3 x]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 8

Raise $\sin (3 x)$ to the power of $1$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- (\sin^{1} (3 x) \sin^{1} (3 x)) \frac{d}{d x} [3 x]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- {\sin (3 x)}^{1 + 1} \frac{d}{d x} [3 x]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 10

Differentiate.

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Step 10.1

Add $1$ and $1$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- \sin^{2} (3 x) \frac{d}{d x} [3 x]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 10.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- \sin^{2} (3 x) (3 \frac{d}{d x} [x])) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 10.3

Multiply $3$ by $- 1$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x) \frac{d}{d x} [x]) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 10.4

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

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x) \cdot 1) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 10.5

Multiply $- 3$ by $1$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + \ln (\cos (3 x)) \frac{d}{d x} [\sin (3 x)])$

Step 11

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 $g (x) = 3 x$ .

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Step 11.1

To apply the Chain Rule, set $u_{4}$ as $3 x$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + \ln (\cos (3 x)) (\frac{d}{d u_{4}} [\sin (u_{4})] \frac{d}{d x} [3 x]))$

Step 11.2

The derivative of $\sin (u_{4})$ with respect to $u_{4}$ is $\cos (u_{4})$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + \ln (\cos (3 x)) (\cos (u_{4}) \frac{d}{d x} [3 x]))$

Step 11.3

Replace all occurrences of $u_{4}$ with $3 x$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + \ln (\cos (3 x)) (\cos (3 x) \frac{d}{d x} [3 x]))$

Step 12

Differentiate.

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Step 12.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + \ln (\cos (3 x)) \cos (3 x) (3 \frac{d}{d x} [x]))$

Step 12.2

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

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + \ln (\cos (3 x)) \cos (3 x) (3 \cdot 1))$

Step 12.3

Simplify the expression.

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Step 12.3.1

Multiply $3$ by $1$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + \ln (\cos (3 x)) \cos (3 x) \cdot 3)$

Step 12.3.2

Move $3$ to the left of $\ln (\cos (3 x)) \cos (3 x)$ .

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x)) + 3 \cdot (\ln (\cos (3 x)) \cos (3 x)))$

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

$e^{\sin (3 x) \ln (\cos (3 x))} (\sec (3 x) (- 3 \sin^{2} (3 x))) + e^{\sin (3 x) \ln (\cos (3 x))} (3 \ln (\cos (3 x)) \cos (3 x))$

Step 13.2

Move $3$ to the left of $e^{\sin (3 x) \ln (\cos (3 x))}$ .

$e^{\sin (3 x) \ln (\cos (3 x))} \sec (3 x) (- 3 \sin^{2} (3 x)) + 3 e^{\sin (3 x) \ln (\cos (3 x))} \ln (\cos (3 x)) \cos (3 x)$

Step 13.3

Reorder terms.

$- 3 e^{\sin (3 x) \ln (\cos (3 x))} \sin^{2} (3 x) \sec (3 x) + 3 e^{\sin (3 x) \ln (\cos (3 x))} \cos (3 x) \ln (\cos (3 x))$