Calculus Examples

$y = \frac{2}{3} \ln (1 + 3 \cos^{2} (x))$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{2}{3} \cdot \ln (1 + 3 \cos^{2} (x)))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $\frac{2}{3}$ is constant with respect to $x$ , the derivative of $\frac{2}{3} \cdot \ln (1 + 3 \cos^{2} (x))$ with respect to $x$ is $\frac{2}{3} \frac{d}{d x} [\ln (1 + 3 \cos^{2} (x))]$ .

$\frac{2}{3} \frac{d}{d x} [\ln (1 + 3 \cos^{2} (x))]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 1 + 3 \cos^{2} (x)$ .

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Step 3.2.1

To apply the Chain Rule, set $u_{1}$ as $1 + 3 \cos^{2} (x)$ .

$\frac{2}{3} (\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d x} [1 + 3 \cos^{2} (x)])$

Step 3.2.2

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

$\frac{2}{3} (\frac{1}{u_{1}} \frac{d}{d x} [1 + 3 \cos^{2} (x)])$

Step 3.2.3

Replace all occurrences of $u_{1}$ with $1 + 3 \cos^{2} (x)$ .

$\frac{2}{3} (\frac{1}{1 + 3 \cos^{2} (x)} \frac{d}{d x} [1 + 3 \cos^{2} (x)])$

Step 3.3

Differentiate.

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Step 3.3.1

Multiply $\frac{1}{1 + 3 \cos^{2} (x)}$ by $\frac{2}{3}$ .

$\frac{2}{(1 + 3 \cos^{2} (x)) \cdot 3} \frac{d}{d x} [1 + 3 \cos^{2} (x)]$

Step 3.3.2

Move $3$ to the left of $1 + 3 \cos^{2} (x)$ .

$\frac{2}{3 \cdot (1 + 3 \cos^{2} (x))} \frac{d}{d x} [1 + 3 \cos^{2} (x)]$

Step 3.3.3

By the Sum Rule, the derivative of $1 + 3 \cos^{2} (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [3 \cos^{2} (x)]$ .

$\frac{2}{3 (1 + 3 \cos^{2} (x))} (\frac{d}{d x} [1] + \frac{d}{d x} [3 \cos^{2} (x)])$

Step 3.3.4

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

$\frac{2}{3 (1 + 3 \cos^{2} (x))} (0 + \frac{d}{d x} [3 \cos^{2} (x)])$

Step 3.3.5

Add $0$ and $\frac{d}{d x} [3 \cos^{2} (x)]$ .

$\frac{2}{3 (1 + 3 \cos^{2} (x))} \frac{d}{d x} [3 \cos^{2} (x)]$

Step 3.3.6

Since $3$ is constant with respect to $x$ , the derivative of $3 \cos^{2} (x)$ with respect to $x$ is $3 \frac{d}{d x} [\cos^{2} (x)]$ .

$\frac{2}{3 (1 + 3 \cos^{2} (x))} (3 \frac{d}{d x} [\cos^{2} (x)])$

Step 3.3.7

Simplify terms.

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Step 3.3.7.1

Combine $3$ and $\frac{2}{3 (1 + 3 \cos^{2} (x))}$ .

$\frac{3 \cdot 2}{3 (1 + 3 \cos^{2} (x))} \frac{d}{d x} [\cos^{2} (x)]$

Step 3.3.7.2

Multiply $3$ by $2$ .

$\frac{6}{3 (1 + 3 \cos^{2} (x))} \frac{d}{d x} [\cos^{2} (x)]$

Step 3.3.7.3

Cancel the common factor of $6$ and $3$ .

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Step 3.3.7.3.1

Factor $3$ out of $6$ .

$\frac{3 \cdot 2}{3 (1 + 3 \cos^{2} (x))} \frac{d}{d x} [\cos^{2} (x)]$

Step 3.3.7.3.2

Cancel the common factors.

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Step 3.3.7.3.2.1

Cancel the common factor.

$\frac{3 \cdot 2}{3 (1 + 3 \cos^{2} (x))} \frac{d}{d x} [\cos^{2} (x)]$

Step 3.3.7.3.2.2

Rewrite the expression.

$\frac{2}{1 + 3 \cos^{2} (x)} \frac{d}{d x} [\cos^{2} (x)]$

Step 3.4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (x)$ .

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Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $\cos (x)$ .

$\frac{2}{1 + 3 \cos^{2} (x)} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d x} [\cos (x)])$

Step 3.4.2

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

$\frac{2}{1 + 3 \cos^{2} (x)} (2 u_{2} \frac{d}{d x} [\cos (x)])$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $\cos (x)$ .

$\frac{2}{1 + 3 \cos^{2} (x)} (2 \cos (x) \frac{d}{d x} [\cos (x)])$

Step 3.5

Combine fractions.

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Step 3.5.1

Combine $2$ and $\frac{2}{1 + 3 \cos^{2} (x)}$ .

$\frac{2 \cdot 2}{1 + 3 \cos^{2} (x)} (\cos (x) \frac{d}{d x} [\cos (x)])$

Step 3.5.2

Multiply $2$ by $2$ .

$\frac{4}{1 + 3 \cos^{2} (x)} (\cos (x) \frac{d}{d x} [\cos (x)])$

Step 3.5.3

Combine $\cos (x)$ and $\frac{4}{1 + 3 \cos^{2} (x)}$ .

$\frac{\cos (x) \cdot 4}{1 + 3 \cos^{2} (x)} \frac{d}{d x} [\cos (x)]$

Step 3.5.4

Move $4$ to the left of $\cos (x)$ .

$\frac{4 \cdot \cos (x)}{1 + 3 \cos^{2} (x)} \frac{d}{d x} [\cos (x)]$

Step 3.6

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

$\frac{4 \cos (x)}{1 + 3 \cos^{2} (x)} (- \sin (x))$

Step 3.7

Combine $\frac{4 \cos (x)}{1 + 3 \cos^{2} (x)}$ and $\sin (x)$ .

$- \frac{4 \cos (x) \sin (x)}{1 + 3 \cos^{2} (x)}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{4 \cos (x) \sin (x)}{1 + 3 \cos^{2} (x)}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{4 \cos (x) \sin (x)}{1 + 3 \cos^{2} (x)}$