Calculus Examples

$y = \sin^{2} (e^{3 x + 1})$

Step 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 $g (x) = \sin (e^{3 x + 1})$ .

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Step 1.1

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

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

Step 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 $n = 2$ .

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (e^{3 x + 1})$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = e^{3 x + 1}$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $e^{3 x + 1}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $e^{3 x + 1}$ .

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

Step 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 $g (x) = 3 x + 1$ .

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Step 3.1

To apply the Chain Rule, set $u_{3}$ as $3 x + 1$ .

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

Step 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 $a$ = $e$ .

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

Step 3.3

Replace all occurrences of $u_{3}$ with $3 x + 1$ .

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

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $3 x + 1$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [1]$ .

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Multiply $3$ by $1$ .

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

Step 4.5

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 \sin (e^{3 x + 1}) \cos (e^{3 x + 1}) (e^{3 x + 1} (3 + 0))$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $3$ and $0$ .

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

Step 4.6.2

Move $3$ to the left of $e^{3 x + 1}$ .

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

Step 4.6.3

Multiply $3$ by $2$ .

$6 \sin (e^{3 x + 1}) \cos (e^{3 x + 1}) e^{3 x + 1}$

Step 4.6.4

Reorder the factors of $6 \sin (e^{3 x + 1}) \cos (e^{3 x + 1}) e^{3 x + 1}$ .

$6 e^{3 x + 1} \cos (e^{3 x + 1}) \sin (e^{3 x + 1})$