Calculus Examples

$y = x^{\sin (x)}$ , $(\frac{π}{2}, \frac{π}{2})$

Step 1

Find the first derivative and evaluate at $x = \frac{π}{2}$ and $y = \frac{π}{2}$ to find the slope of the tangent line.

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Step 1.1

Use the properties of logarithms to simplify the differentiation.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

To apply the Chain Rule, set $u$ as $\sin (x) \ln (x)$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $\sin (x) \ln (x)$ .

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

Step 1.3

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 $g (x) = \ln (x)$ .

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

Step 1.4

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

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

Step 1.5

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

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

Step 1.6

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

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

Step 1.7

Simplify.

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Step 1.7.1

Apply the distributive property.

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

Step 1.7.2

Combine $e^{\sin (x) \ln (x)}$ and $\frac{\sin (x)}{x}$ .

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

Step 1.7.3

Reorder terms.

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

Step 1.8

Evaluate the derivative at $x = \frac{π}{2}$ .

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

Step 1.9

Simplify.

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

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

Step 1.9.1.2

Simplify $1 \ln (\frac{π}{2})$ by moving $1$ inside the logarithm.

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

Step 1.9.1.3

Exponentiation and log are inverse functions.

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

Step 1.9.1.4

Simplify.

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

Step 1.9.1.5

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$\frac{π}{2} \cdot 0 \ln (\frac{π}{2}) + \frac{e^{\sin (\frac{π}{2}) \ln (\frac{π}{2})} \sin (\frac{π}{2})}{\frac{π}{2}}$

Step 1.9.1.6

Multiply $\frac{π}{2}$ by $0$ .

$0 \ln (\frac{π}{2}) + \frac{e^{\sin (\frac{π}{2}) \ln (\frac{π}{2})} \sin (\frac{π}{2})}{\frac{π}{2}}$

Step 1.9.1.7

Multiply $0$ by $\ln (\frac{π}{2})$ .

$0 + \frac{e^{\sin (\frac{π}{2}) \ln (\frac{π}{2})} \sin (\frac{π}{2})}{\frac{π}{2}}$

Step 1.9.1.8

Multiply the numerator by the reciprocal of the denominator.

$0 + e^{\sin (\frac{π}{2}) \ln (\frac{π}{2})} \sin (\frac{π}{2}) \frac{2}{π}$

Step 1.9.1.9

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 + e^{1 \ln (\frac{π}{2})} \sin (\frac{π}{2}) \frac{2}{π}$

Step 1.9.1.10

Simplify $1 \ln (\frac{π}{2})$ by moving $1$ inside the logarithm.

$0 + e^{\ln ({(\frac{π}{2})}^{1})} \sin (\frac{π}{2}) \frac{2}{π}$

Step 1.9.1.11

Exponentiation and log are inverse functions.

$0 + {(\frac{π}{2})}^{1} \sin (\frac{π}{2}) \frac{2}{π}$

Step 1.9.1.12

Simplify.

$0 + \frac{π}{2} \sin (\frac{π}{2}) \frac{2}{π}$

Step 1.9.1.13

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$0 + \frac{π}{2} \cdot 1 \frac{2}{π}$

Step 1.9.1.14

Multiply $\frac{π}{2}$ by $1$ .

$0 + \frac{π}{2} \cdot \frac{2}{π}$

Step 1.9.1.15

Cancel the common factor of $π$ .

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Step 1.9.1.15.1

Cancel the common factor.

$0 + \frac{π}{2} \cdot \frac{2}{π}$

Step 1.9.1.15.2

Rewrite the expression.

$0 + \frac{1}{2} \cdot 2$

Step 1.9.1.16

Cancel the common factor of $2$ .

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Step 1.9.1.16.1

Cancel the common factor.

$0 + \frac{1}{2} \cdot 2$

Step 1.9.1.16.2

Rewrite the expression.

$0 + 1$

Step 1.9.2

Add $0$ and $1$ .

$1$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $1$ and a given point $(\frac{π}{2}, \frac{π}{2})$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (\frac{π}{2}) = 1 \cdot (x - (\frac{π}{2}))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - \frac{π}{2} = 1 \cdot (x - \frac{π}{2})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Multiply $x - \frac{π}{2}$ by $1$ .

$y - \frac{π}{2} = x - \frac{π}{2}$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$y = x - \frac{π}{2} + \frac{π}{2}$

Step 2.3.2.2

Combine the opposite terms in $x - \frac{π}{2} + \frac{π}{2}$ .

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Step 2.3.2.2.1

Add $- \frac{π}{2}$ and $\frac{π}{2}$ .

$y = x + 0$

Step 2.3.2.2.2

Add $x$ and $0$ .

$y = x$

Step 3