Calculus Examples

$y = x^{2} \cos (x)$

Step 1

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

$\ln (y) = \ln (x^{2} \cos (x))$

Step 2

Expand the right hand side.

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Step 2.1

Rewrite $\ln (x^{2} \cos (x))$ as $\ln (x^{2}) + \ln (\cos (x))$ .

$\ln (y) = \ln (x^{2}) + \ln (\cos (x))$

Step 2.2

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

$\ln (y) = 2 \ln (x) + \ln (\cos (x))$

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.

$\frac{y'}{y} = 2 \ln (x) + \ln (\cos (x))$

Step 3.2

Differentiate the right hand side.

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Step 3.2.1

Differentiate $2 \ln (x) + \ln (\cos (x))$ .

$\frac{y'}{y} = \frac{d}{d x} [2 \ln (x) + \ln (\cos (x))]$

Step 3.2.2

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

$\frac{y'}{y} = \frac{d}{d x} [2 \ln (x)] + \frac{d}{d x} [\ln (\cos (x))]$

Step 3.2.3

Evaluate $\frac{d}{d x} [2 \ln (x)]$ .

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Step 3.2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 \ln (x)$ with respect to $x$ is $2 \frac{d}{d x} [\ln (x)]$ .

$\frac{y'}{y} = 2 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [\ln (\cos (x))]$

Step 3.2.3.2

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

$\frac{y'}{y} = 2 \frac{1}{x} + \frac{d}{d x} [\ln (\cos (x))]$

Step 3.2.3.3

Combine $2$ and $\frac{1}{x}$ .

$\frac{y'}{y} = \frac{2}{x} + \frac{d}{d x} [\ln (\cos (x))]$

Step 3.2.4

Evaluate $\frac{d}{d x} [\ln (\cos (x))]$ .

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Step 3.2.4.1

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

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Step 3.2.4.1.1

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

$\frac{y'}{y} = \frac{2}{x} + \frac{d}{d u} [\ln (u)] \frac{d}{d x} [\cos (x)]$

Step 3.2.4.1.2

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

$\frac{y'}{y} = \frac{2}{x} + \frac{1}{u} \frac{d}{d x} [\cos (x)]$

Step 3.2.4.1.3

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

$\frac{y'}{y} = \frac{2}{x} + \frac{1}{\cos (x)} \frac{d}{d x} [\cos (x)]$

Step 3.2.4.2

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

$\frac{y'}{y} = \frac{2}{x} + \frac{1}{\cos (x)} (- \sin (x))$

Step 3.2.4.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{y'}{y} = \frac{2}{x} + \sec (x) (- \sin (x))$

Step 3.2.5

Simplify.

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Step 3.2.5.1

Reorder terms.

$\frac{y'}{y} = - \sec (x) \sin (x) + \frac{2}{x}$

Step 3.2.5.2

Simplify each term.

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Step 3.2.5.2.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{y'}{y} = - \frac{1}{\cos (x)} \sin (x) + \frac{2}{x}$

Step 3.2.5.2.2

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

$\frac{y'}{y} = - \frac{\sin (x)}{\cos (x)} + \frac{2}{x}$

Step 3.2.5.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{y'}{y} = - \tan (x) + \frac{2}{x}$

Step 4

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

$y' = (- \tan (x) + \frac{2}{x}) (x^{2} \cos (x))$

Step 5

Simplify the right hand side.

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Step 5.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$y' = (- \frac{\sin (x)}{\cos (x)} + \frac{2}{x}) (x^{2} \cos (x))$

Step 5.2

Apply the distributive property.

$y' = - \frac{\sin (x)}{\cos (x)} (x^{2} \cos (x)) + \frac{2}{x} (x^{2} \cos (x))$

Step 5.3

Cancel the common factor of $\cos (x)$ .

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Step 5.3.1

Move the leading negative in $- \frac{\sin (x)}{\cos (x)}$ into the numerator.

$y' = \frac{- \sin (x)}{\cos (x)} (x^{2} \cos (x)) + \frac{2}{x} (x^{2} \cos (x))$

Step 5.3.2

Factor $\cos (x)$ out of $x^{2} \cos (x)$ .

$y' = \frac{- \sin (x)}{\cos (x)} (\cos (x) x^{2}) + \frac{2}{x} (x^{2} \cos (x))$

Step 5.3.3

Cancel the common factor.

$y' = \frac{- \sin (x)}{\cos (x)} (\cos (x) x^{2}) + \frac{2}{x} (x^{2} \cos (x))$

Step 5.3.4

Rewrite the expression.

$y' = - \sin (x) x^{2} + \frac{2}{x} (x^{2} \cos (x))$

Step 5.4

Cancel the common factor of $x$ .

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Step 5.4.1

Factor $x$ out of $x^{2} \cos (x)$ .

$y' = - \sin (x) x^{2} + \frac{2}{x} (x (x \cos (x)))$

Step 5.4.2

Cancel the common factor.

$y' = - \sin (x) x^{2} + \frac{2}{x} (x (x \cos (x)))$

Step 5.4.3

Rewrite the expression.

$y' = - \sin (x) x^{2} + 2 (x \cos (x))$

Step 5.5

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

$y' = - x^{2} \sin (x) + 2 x \cos (x)$