Calculus Examples

$f (x) = \cos (\frac{3 x + 1}{2 x - 1})$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{3 x + 1}{2 x - 1}$ .

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

Step 1.2

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

$- \sin (u) \frac{d}{d x} [\frac{3 x + 1}{2 x - 1}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{3 x + 1}{2 x - 1}$ .

$- \sin (\frac{3 x + 1}{2 x - 1}) \frac{d}{d x} [\frac{3 x + 1}{2 x - 1}]$

Step 2

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 $f (x) = 3 x + 1$ and $g (x) = 2 x - 1$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $3$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

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

Step 3.6.2

Move $3$ to the left of $2 x - 1$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Multiply $2$ by $1$ .

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

Step 3.11

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

$- \sin (\frac{3 x + 1}{2 x - 1}) \frac{3 (2 x - 1) - (3 x + 1) (2 + 0)}{{(2 x - 1)}^{2}}$

Step 3.12

Combine fractions.

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Step 3.12.1

Add $2$ and $0$ .

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

Step 3.12.2

Multiply $2$ by $- 1$ .

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

Step 3.12.3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Simplify the numerator.

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Step 4.3.1

Combine the opposite terms in $3 (2 x) + 3 \cdot - 1 - 2 (3 x) - 2 \cdot 1$ .

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Step 4.3.1.1

Reorder the factors in the terms $3 (2 x)$ and $- 2 (3 x)$ .

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

Step 4.3.1.2

Subtract $2 \cdot 3 x$ from $2 \cdot 3 x$ .

$- \frac{(3 \cdot - 1 + 0 - 2 \cdot 1) \sin (\frac{3 x + 1}{2 x - 1})}{{(2 x - 1)}^{2}}$

Step 4.3.1.3

Add $3 \cdot - 1$ and $0$ .

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

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Multiply $3$ by $- 1$ .

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

Step 4.3.2.2

Multiply $- 2$ by $1$ .

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

Step 4.3.3

Subtract $2$ from $- 3$ .

$- \frac{- 5 \sin (\frac{3 x + 1}{2 x - 1})}{{(2 x - 1)}^{2}}$

Step 4.4

Combine terms.

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Step 4.4.1

Move the negative in front of the fraction.

$- - \frac{5 \sin (\frac{3 x + 1}{2 x - 1})}{{(2 x - 1)}^{2}}$

Step 4.4.2

Multiply $- 1$ by $- 1$ .

$1 \frac{5 \sin (\frac{3 x + 1}{2 x - 1})}{{(2 x - 1)}^{2}}$

Step 4.4.3

Multiply $\frac{5 \sin (\frac{3 x + 1}{2 x - 1})}{{(2 x - 1)}^{2}}$ by $1$ .

$\frac{5 \sin (\frac{3 x + 1}{2 x - 1})}{{(2 x - 1)}^{2}}$