Calculus Examples

$y = 6 x - 4 \cos (3 x)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (6 x - 4 \cos (3 x))$

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

By the Sum Rule, the derivative of $6 x - 4 \cos (3 x)$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 4 \cos (3 x)]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.2

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 3.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.2.2

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

$6 \cdot 1 + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.2.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 4 \cos (3 x)]$

Step 3.3

Evaluate $\frac{d}{d x} [- 4 \cos (3 x)]$ .

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Step 3.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 \cos (3 x)$ with respect to $x$ is $- 4 \frac{d}{d x} [\cos (3 x)]$ .

$6 - 4 \frac{d}{d x} [\cos (3 x)]$

Step 3.3.2

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

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Step 3.3.2.1

To apply the Chain Rule, set $u$ as $3 x$ .

$6 - 4 (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [3 x])$

Step 3.3.2.2

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

$6 - 4 (- \sin (u) \frac{d}{d x} [3 x])$

Step 3.3.2.3

Replace all occurrences of $u$ with $3 x$ .

$6 - 4 (- \sin (3 x) \frac{d}{d x} [3 x])$

Step 3.3.3

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

$6 - 4 (- \sin (3 x) (3 \frac{d}{d x} [x]))$

Step 3.3.4

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

$6 - 4 (- \sin (3 x) (3 \cdot 1))$

Step 3.3.5

Multiply $3$ by $1$ .

$6 - 4 (- \sin (3 x) \cdot 3)$

Step 3.3.6

Multiply $3$ by $- 1$ .

$6 - 4 (- 3 \sin (3 x))$

Step 3.3.7

Multiply $- 3$ by $- 4$ .

$6 + 12 \sin (3 x)$

Step 4

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

$y' = 6 + 12 \sin (3 x)$

Step 5

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

$\frac{d y}{d x} = 6 + 12 \sin (3 x)$