Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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 $f (x) = \ln (x)$ and $g (x) = \cos (x)$ .

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

Step 1.2

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

$\ln (x) (- \sin (x)) + \cos (x) \frac{d}{d x} [\ln (x)]$

Step 1.3

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

$\ln (x) (- \sin (x)) + \cos (x) \frac{1}{x}$

Step 1.4

Combine fractions.

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Step 1.4.1

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

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

Step 1.4.2

Reorder terms.

$f' (x) = - \sin (x) \ln (x) + \frac{\cos (x)}{x}$

Step 2

Find the second derivative.

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Step 2.1

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

$\frac{d}{d x} [- \sin (x) \ln (x)] + \frac{d}{d x} [\frac{\cos (x)}{x}]$

Step 2.2

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

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Step 2.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- \sin (x) \ln (x)$ with respect to $x$ is $- \frac{d}{d x} [\sin (x) \ln (x)]$ .

$- \frac{d}{d x} [\sin (x) \ln (x)] + \frac{d}{d x} [\frac{\cos (x)}{x}]$

Step 2.2.2

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

$- (\sin (x) \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [\frac{\cos (x)}{x}]$

Step 2.2.3

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

$- (\sin (x) \frac{1}{x} + \ln (x) \frac{d}{d x} [\sin (x)]) + \frac{d}{d x} [\frac{\cos (x)}{x}]$

Step 2.2.4

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

$- (\sin (x) \frac{1}{x} + \ln (x) \cos (x)) + \frac{d}{d x} [\frac{\cos (x)}{x}]$

Step 2.2.5

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

$- (\frac{\sin (x)}{x} + \ln (x) \cos (x)) + \frac{d}{d x} [\frac{\cos (x)}{x}]$

Step 2.3

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

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Step 2.3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \cos (x)$ and $g (x) = x$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $- 1$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Combine terms.

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Step 2.4.2.1

To write $- \frac{\sin (x)}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 2.4.2.2

Write each expression with a common denominator of $x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.2.2.1

Multiply $\frac{\sin (x)}{x}$ by $\frac{x}{x}$ .

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

Step 2.4.2.2.2

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

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

Step 2.4.2.2.3

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

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

Step 2.4.2.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.4.2.2.5

Add $1$ and $1$ .

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

Step 2.4.2.3

Combine the numerators over the common denominator.

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

Step 2.4.2.4

Reorder $x$ and $- 1$ .

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

Step 2.4.2.5

Rewrite $- 1 x$ as $- x$ .

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

Step 2.4.2.6

Subtract $x \sin (x)$ from $- x \sin (x)$ .

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

Step 2.4.2.7

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

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

Step 2.4.2.8

Combine $- \ln (x) \cos (x)$ and $\frac{x^{2}}{x^{2}}$ .

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

Step 2.4.2.9

Combine the numerators over the common denominator.

$\frac{- \ln (x) \cos (x) x^{2} - 2 x \sin (x) - \cos (x)}{x^{2}}$

Step 2.4.3

Reorder terms.

$\frac{- x^{2} \cos (x) \ln (x) - 2 x \sin (x) - \cos (x)}{x^{2}}$

Step 2.4.4

Factor $- 1$ out of $- x^{2} \cos (x) \ln (x)$ .

$\frac{- (x^{2} \cos (x) \ln (x)) - 2 x \sin (x) - \cos (x)}{x^{2}}$

Step 2.4.5

Factor $- 1$ out of $- 2 x \sin (x)$ .

$\frac{- (x^{2} \cos (x) \ln (x)) - (2 x \sin (x)) - \cos (x)}{x^{2}}$

Step 2.4.6

Factor $- 1$ out of $- (x^{2} \cos (x) \ln (x)) - (2 x \sin (x))$ .

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

Step 2.4.7

Factor $- 1$ out of $- \cos (x)$ .

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

Step 2.4.8

Factor $- 1$ out of $- (x^{2} \cos (x) \ln (x) + 2 x \sin (x)) - (\cos (x))$ .

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

Step 2.4.9

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

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

Step 2.4.10

Move the negative in front of the fraction.

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