Calculus Examples

$y = \sin^{2} (e^{5 x}) + \cos^{2} (e^{5 x})$

Step 1

By the Sum Rule, the derivative of $\sin^{2} (e^{5 x}) + \cos^{2} (e^{5 x})$ with respect to $x$ is $\frac{d}{d x} [\sin^{2} (e^{5 x})] + \frac{d}{d x} [\cos^{2} (e^{5 x})]$ .

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

Step 2

Evaluate $\frac{d}{d x} [\sin^{2} (e^{5 x})]$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (e^{5 x})$ .

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Step 2.1.1

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

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

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

Step 2.1.3

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

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

Step 2.2

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

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Step 2.2.1

To apply the Chain Rule, set $u_{2}$ as $e^{5 x}$ .

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

Step 2.2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

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

Step 2.2.3

Replace all occurrences of $u_{2}$ with $e^{5 x}$ .

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

Step 2.3

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 $g (x) = 5 x$ .

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Step 2.3.1

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

$2 \sin (e^{5 x}) (\cos (e^{5 x}) (\frac{d}{d u_{3}} [e^{u_{3}}] \frac{d}{d x} [5 x])) + \frac{d}{d x} [\cos^{2} (e^{5 x})]$

Step 2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{3}} [a^{u_{3}}]$ is $a^{u_{3}} \ln (a)$ where $a$ = $e$ .

$2 \sin (e^{5 x}) (\cos (e^{5 x}) (e^{u_{3}} \frac{d}{d x} [5 x])) + \frac{d}{d x} [\cos^{2} (e^{5 x})]$

Step 2.3.3

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

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

Step 2.4

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

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

Step 2.5

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

$2 \sin (e^{5 x}) (\cos (e^{5 x}) (e^{5 x} (5 \cdot 1))) + \frac{d}{d x} [\cos^{2} (e^{5 x})]$

Step 2.6

Multiply $5$ by $1$ .

$2 \sin (e^{5 x}) (\cos (e^{5 x}) (e^{5 x} \cdot 5)) + \frac{d}{d x} [\cos^{2} (e^{5 x})]$

Step 2.7

Move $5$ to the left of $e^{5 x}$ .

$2 \sin (e^{5 x}) (\cos (e^{5 x}) (5 \cdot e^{5 x})) + \frac{d}{d x} [\cos^{2} (e^{5 x})]$

Step 2.8

Multiply $5$ by $2$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + \frac{d}{d x} [\cos^{2} (e^{5 x})]$

Step 3

Evaluate $\frac{d}{d x} [\cos^{2} (e^{5 x})]$ .

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (e^{5 x})$ .

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Step 3.1.1

To apply the Chain Rule, set $u_{4}$ as $\cos (e^{5 x})$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + \frac{d}{d u_{4}} [{u_{4}}^{2}] \frac{d}{d x} [\cos (e^{5 x})]$

Step 3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{4}} [{u_{4}}^{n}]$ is $n {u_{4}}^{n - 1}$ where $n = 2$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 u_{4} \frac{d}{d x} [\cos (e^{5 x})]$

Step 3.1.3

Replace all occurrences of $u_{4}$ with $\cos (e^{5 x})$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) \frac{d}{d x} [\cos (e^{5 x})]$

Step 3.2

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

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Step 3.2.1

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

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (\frac{d}{d u_{5}} [\cos (u_{5})] \frac{d}{d x} [e^{5 x}])$

Step 3.2.2

The derivative of $\cos (u_{5})$ with respect to $u_{5}$ is $- \sin (u_{5})$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (u_{5}) \frac{d}{d x} [e^{5 x}])$

Step 3.2.3

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

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) \frac{d}{d x} [e^{5 x}])$

Step 3.3

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 $g (x) = 5 x$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{6}$ as $5 x$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) (\frac{d}{d u_{6}} [e^{u_{6}}] \frac{d}{d x} [5 x]))$

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{6}} [a^{u_{6}}]$ is $a^{u_{6}} \ln (a)$ where $a$ = $e$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) (e^{u_{6}} \frac{d}{d x} [5 x]))$

Step 3.3.3

Replace all occurrences of $u_{6}$ with $5 x$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) (e^{5 x} \frac{d}{d x} [5 x]))$

Step 3.4

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

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) (e^{5 x} (5 \frac{d}{d x} [x])))$

Step 3.5

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

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) (e^{5 x} (5 \cdot 1)))$

Step 3.6

Multiply $5$ by $1$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) (e^{5 x} \cdot 5))$

Step 3.7

Move $5$ to the left of $e^{5 x}$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- \sin (e^{5 x}) (5 \cdot e^{5 x}))$

Step 3.8

Multiply $5$ by $- 1$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} + 2 \cos (e^{5 x}) (- 5 \sin (e^{5 x}) e^{5 x})$

Step 3.9

Multiply $- 5$ by $2$ .

$10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x} - 10 \cos (e^{5 x}) \sin (e^{5 x}) e^{5 x}$

Step 4

Combine terms.

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Step 4.1

Reorder the factors of $10 \sin (e^{5 x}) \cos (e^{5 x}) e^{5 x}$ .

$10 e^{5 x} \cos (e^{5 x}) \sin (e^{5 x}) - 10 \cos (e^{5 x}) \sin (e^{5 x}) e^{5 x}$

Step 4.2

Subtract $10 \cos (e^{5 x}) \sin (e^{5 x}) e^{5 x}$ from $10 e^{5 x} \cos (e^{5 x}) \sin (e^{5 x})$ .

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Step 4.2.1

Move $\cos (e^{5 x})$ .

$10 e^{5 x} \cos (e^{5 x}) \sin (e^{5 x}) - 10 e^{5 x} \cos (e^{5 x}) \sin (e^{5 x})$

Step 4.2.2

Subtract $10 e^{5 x} \cos (e^{5 x}) \sin (e^{5 x})$ from $10 e^{5 x} \cos (e^{5 x}) \sin (e^{5 x})$ .

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