Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1

Rewrite ${(1 + x^{2})}^{\sin (x)}$ as $e^{\ln ({(1 + x^{2})}^{\sin (x)})}$ .

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

Step 1.2

Expand $\ln ({(1 + x^{2})}^{\sin (x)})$ by moving $\sin (x)$ outside the logarithm.

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

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 $g (x) = \sin (x) \ln (1 + x^{2})$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x) \ln (1 + x^{2})$ .

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

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 $a$ = $e$ .

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

Step 2.3

Replace all occurrences of $u_{1}$ with $\sin (x) \ln (1 + x^{2})$ .

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

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 $f (x) = \sin (x)$ and $g (x) = \ln (1 + x^{2})$ .

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

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 $g (x) = 1 + x^{2}$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate.

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Step 5.1

Combine $\frac{1}{1 + x^{2}}$ and $\sin (x)$ .

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

Step 5.2

By the Sum Rule, the derivative of $1 + x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{2}]$ .

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

Step 5.3

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

$e^{\sin (x) \ln (1 + x^{2})} (\frac{\sin (x)}{1 + x^{2}} (0 + \frac{d}{d x} [x^{2}]) + \ln (1 + x^{2}) \frac{d}{d x} [\sin (x)])$

Step 5.4

Add $0$ and $\frac{d}{d x} [x^{2}]$ .

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

Step 5.5

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

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

Step 5.6

Combine fractions.

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Step 5.6.1

Combine $2$ and $\frac{\sin (x)}{1 + x^{2}}$ .

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

Step 5.6.2

Combine $\frac{2 \sin (x)}{1 + x^{2}}$ and $x$ .

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

Step 6

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$e^{\sin (x) \ln (1 + x^{2})} (\frac{2 \sin (x) x}{1 + x^{2}} + \ln (1 + x^{2}) \cos (x))$

Step 7

To write $\ln (1 + x^{2}) \cos (x)$ as a fraction with a common denominator, multiply by $\frac{1 + x^{2}}{1 + x^{2}}$ .

$e^{\sin (x) \ln (1 + x^{2})} (\frac{2 \sin (x) x}{1 + x^{2}} + \frac{\ln (1 + x^{2}) \cos (x) (1 + x^{2})}{1 + x^{2}})$

Step 8

Combine the numerators over the common denominator.

$e^{\sin (x) \ln (1 + x^{2})} \frac{2 \sin (x) x + \ln (1 + x^{2}) \cos (x) (1 + x^{2})}{1 + x^{2}}$

Step 9

Combine $e^{\sin (x) \ln (1 + x^{2})}$ and $\frac{2 \sin (x) x + \ln (1 + x^{2}) \cos (x) (1 + x^{2})}{1 + x^{2}}$ .

$\frac{e^{\sin (x) \ln (1 + x^{2})} (2 \sin (x) x + \ln (1 + x^{2}) \cos (x) (1 + x^{2}))}{1 + x^{2}}$

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Simplify each term.

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Step 10.1.1.1

Apply the distributive property.

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

Step 10.1.1.2

Multiply $\ln (1 + x^{2})$ by $1$ .

$\frac{e^{\sin (x) \ln (1 + x^{2})} (2 \sin (x) x + \ln (1 + x^{2}) \cos (x) + \ln (1 + x^{2}) \cos (x) x^{2})}{1 + x^{2}}$

Step 10.1.2

Apply the distributive property.

$\frac{e^{\sin (x) \ln (1 + x^{2})} (2 \sin (x) x) + e^{\sin (x) \ln (1 + x^{2})} (\ln (1 + x^{2}) \cos (x)) + e^{\sin (x) \ln (1 + x^{2})} (\ln (1 + x^{2}) \cos (x) x^{2})}{1 + x^{2}}$

Step 10.1.3

Rewrite using the commutative property of multiplication.

$\frac{2 e^{\sin (x) \ln (1 + x^{2})} \sin (x) x + e^{\sin (x) \ln (1 + x^{2})} \ln (1 + x^{2}) \cos (x) + e^{\sin (x) \ln (1 + x^{2})} \ln (1 + x^{2}) \cos (x) x^{2}}{1 + x^{2}}$

Step 10.1.4

Reorder factors in $2 e^{\sin (x) \ln (1 + x^{2})} \sin (x) x + e^{\sin (x) \ln (1 + x^{2})} \ln (1 + x^{2}) \cos (x) + e^{\sin (x) \ln (1 + x^{2})} \ln (1 + x^{2}) \cos (x) x^{2}$ .

$\frac{2 x e^{\sin (x) \ln (1 + x^{2})} \sin (x) + e^{\sin (x) \ln (1 + x^{2})} \ln (1 + x^{2}) \cos (x) + x^{2} e^{\sin (x) \ln (1 + x^{2})} \ln (1 + x^{2}) \cos (x)}{1 + x^{2}}$

Step 10.2

Reorder terms.

$\frac{2 e^{\sin (x) \ln (1 + x^{2})} x \sin (x) + e^{\sin (x) \ln (1 + x^{2})} \cos (x) \ln (1 + x^{2}) + e^{\sin (x) \ln (1 + x^{2})} x^{2} \cos (x) \ln (1 + x^{2})}{x^{2} + 1}$