Calculus Examples

$y = \frac{5}{{(2 x)}^{3}} + 2 \cos (x)$

Step 1

By the Sum Rule, the derivative of $\frac{5}{{(2 x)}^{3}} + 2 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{{(2 x)}^{3}}] + \frac{d}{d x} [2 \cos (x)]$ .

$\frac{d}{d x} [\frac{5}{{(2 x)}^{3}}] + \frac{d}{d x} [2 \cos (x)]$

Step 2

Evaluate $\frac{d}{d x} [\frac{5}{{(2 x)}^{3}}]$ .

Tap for more steps...

Step 2.1

Factor $2$ out of $2 x$ .

$\frac{d}{d x} [\frac{5}{{(2 (x))}^{3}}] + \frac{d}{d x} [2 \cos (x)]$

Step 2.2

Apply the product rule to $2 (x)$ .

$\frac{d}{d x} [\frac{5}{2^{3} x^{3}}] + \frac{d}{d x} [2 \cos (x)]$

Step 2.3

Raise $2$ to the power of $3$ .

$\frac{d}{d x} [\frac{5}{8 x^{3}}] + \frac{d}{d x} [2 \cos (x)]$

Step 2.4

Since $\frac{5}{8}$ is constant with respect to $x$ , the derivative of $\frac{5}{8 x^{3}}$ with respect to $x$ is $\frac{5}{8} \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$\frac{5}{8} \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [2 \cos (x)]$

Step 2.5

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$\frac{5}{8} \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [2 \cos (x)]$

Step 2.6

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

Tap for more steps...

Step 2.6.1

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

$\frac{5}{8} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [2 \cos (x)]$

Step 2.6.2

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

$\frac{5}{8} (- u^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [2 \cos (x)]$

Step 2.6.3

Replace all occurrences of $u$ with $x^{3}$ .

$\frac{5}{8} (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [2 \cos (x)]$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{5}{8} (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [2 \cos (x)]$

Step 2.8

Multiply the exponents in ${(x^{3})}^{- 2}$ .

Tap for more steps...

Step 2.8.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{8} (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [2 \cos (x)]$

Step 2.8.2

Multiply $3$ by $- 2$ .

$\frac{5}{8} (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [2 \cos (x)]$

Step 2.9

Multiply $3$ by $- 1$ .

$\frac{5}{8} (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [2 \cos (x)]$

Step 2.10

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 2.10.1

Move $x^{2}$ .

$\frac{5}{8} (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [2 \cos (x)]$

Step 2.10.2

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

$\frac{5}{8} (- 3 x^{2 - 6}) + \frac{d}{d x} [2 \cos (x)]$

Step 2.10.3

Subtract $6$ from $2$ .

$\frac{5}{8} (- 3 x^{- 4}) + \frac{d}{d x} [2 \cos (x)]$

Step 2.11

Combine $- 3$ and $\frac{5}{8}$ .

$\frac{- 3 \cdot 5}{8} x^{- 4} + \frac{d}{d x} [2 \cos (x)]$

Step 2.12

Multiply $- 3$ by $5$ .

$\frac{- 15}{8} x^{- 4} + \frac{d}{d x} [2 \cos (x)]$

Step 2.13

Combine $\frac{- 15}{8}$ and $x^{- 4}$ .

$\frac{- 15 x^{- 4}}{8} + \frac{d}{d x} [2 \cos (x)]$

Step 2.14

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 15}{8 x^{4}} + \frac{d}{d x} [2 \cos (x)]$

Step 2.15

Move the negative in front of the fraction.

$- \frac{15}{8 x^{4}} + \frac{d}{d x} [2 \cos (x)]$

Step 3

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

Tap for more steps...

Step 3.1

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

$- \frac{15}{8 x^{4}} + 2 \frac{d}{d x} [\cos (x)]$

Step 3.2

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

$- \frac{15}{8 x^{4}} + 2 (- \sin (x))$

Step 3.3

Multiply $- 1$ by $2$ .

$- \frac{15}{8 x^{4}} - 2 \sin (x)$