Calculus Examples

$y = x^{5 \sin (x)}$

Step 1

Let $y = f (x)$ , take the natural logarithm of both sides $\ln (y) = \ln (f (x))$ .

$\ln (y) = \ln (x^{5 \sin (x)})$

Step 2

Expand $\ln (x^{5 \sin (x)})$ by moving $5 \sin (x)$ outside the logarithm.

$\ln (y) = 5 \sin (x) \ln (x)$

Step 3

Differentiate the expression using the chain rule, keeping in mind that $y$ is a function of $x$ .

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Step 3.1

Differentiate the left hand side $\ln (y)$ using the chain rule.

$\frac{y'}{y} = 5 \sin (x) \ln (x)$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $5 \sin (x) \ln (x)$ .

$\frac{y'}{y} = \frac{d}{d x} [5 \sin (x) \ln (x)]$

Step 3.2.2

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

$\frac{y'}{y} = 5 \frac{d}{d x} [\sin (x) \ln (x)]$

Step 3.2.3

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

$\frac{y'}{y} = 5 (\sin (x) \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [\sin (x)])$

Step 3.2.4

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

$\frac{y'}{y} = 5 (\sin (x) \frac{1}{x} + \ln (x) \frac{d}{d x} [\sin (x)])$

Step 3.2.5

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

$\frac{y'}{y} = 5 (\frac{\sin (x)}{x} + \ln (x) \frac{d}{d x} [\sin (x)])$

Step 3.2.6

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

$\frac{y'}{y} = 5 (\frac{\sin (x)}{x} + \ln (x) \cos (x))$

Step 3.2.7

Simplify.

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Step 3.2.7.1

Apply the distributive property.

$\frac{y'}{y} = 5 \frac{\sin (x)}{x} + 5 (\ln (x) \cos (x))$

Step 3.2.7.2

Combine $5$ and $\frac{\sin (x)}{x}$ .

$\frac{y'}{y} = \frac{5 \sin (x)}{x} + 5 \ln (x) \cos (x)$

Step 3.2.7.3

Reorder terms.

$\frac{y'}{y} = 5 \cos (x) \ln (x) + \frac{5 \sin (x)}{x}$

Step 4

Isolate $y'$ and substitute the original function for $y$ in the right hand side.

$y' = (5 \cos (x) \ln (x) + \frac{5 \sin (x)}{x}) x^{5 \sin (x)}$

Step 5

Simplify the right hand side.

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Step 5.1

Multiply $5 \cos (x) \ln (x)$ .

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Step 5.1.1

Reorder $\ln (x)$ and $5$ .

$y' = (5 \cdot \ln (x) \cos (x) + \frac{5 \sin (x)}{x}) x^{5 \sin (x)}$

Step 5.1.2

Simplify $5 \cdot \ln (x)$ by moving $5$ inside the logarithm.

$y' = (\ln (x^{5}) \cos (x) + \frac{5 \sin (x)}{x}) x^{5 \sin (x)}$

Step 5.2

Apply the distributive property.

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + \frac{5 \sin (x)}{x} x^{5 \sin (x)}$

Step 5.3

Combine $\frac{5 \sin (x)}{x}$ and $x^{5 \sin (x)}$ .

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + \frac{5 \sin (x) x^{5 \sin (x)}}{x}$

Step 5.4

Cancel the common factor of $x^{5 \sin (x)}$ and $x$ .

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Step 5.4.1

Factor $x$ out of $5 \sin (x) x^{5 \sin (x)}$ .

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + \frac{x (5 \sin (x) x^{5 \sin (x) - 1})}{x}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Raise $x$ to the power of $1$ .

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + \frac{x (5 \sin (x) x^{5 \sin (x) - 1})}{x^{1}}$

Step 5.4.2.2

Factor $x$ out of $x^{1}$ .

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + \frac{x (5 \sin (x) x^{5 \sin (x) - 1})}{x \cdot 1}$

Step 5.4.2.3

Cancel the common factor.

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + \frac{x (5 \sin (x) x^{5 \sin (x) - 1})}{x \cdot 1}$

Step 5.4.2.4

Rewrite the expression.

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + \frac{5 \sin (x) x^{5 \sin (x) - 1}}{1}$

Step 5.4.2.5

Divide $5 \sin (x) x^{5 \sin (x) - 1}$ by $1$ .

$y' = \ln (x^{5}) \cos (x) x^{5 \sin (x)} + 5 \sin (x) x^{5 \sin (x) - 1}$

Step 5.5

Reorder factors in $\ln (x^{5}) \cos (x) x^{5 \sin (x)} + 5 \sin (x) x^{5 \sin (x) - 1}$ .

$y' = x^{5 \sin (x)} \ln (x^{5}) \cos (x) + 5 x^{5 \sin (x) - 1} \sin (x)$