Calculus Examples

$f (x) = 2 x^{3} \cdot (6 x) - \frac{1}{x} + 3 e^{x} - \sin (x)$

Step 1

By the Sum Rule, the derivative of $2 x^{3} \cdot (6 x) - \frac{1}{x} + 3 e^{x} - \sin (x)$ with respect to $x$ is $\frac{d}{d x} [2 x^{3} \cdot (6 x)] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$ .

$\frac{d}{d x} [2 x^{3} \cdot (6 x)] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 2

Evaluate $\frac{d}{d x} [2 x^{3} \cdot (6 x)]$ .

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Step 2.1

Multiply $6$ by $2$ .

$\frac{d}{d x} [12 x^{3} \cdot x] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 2.2

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

$\frac{d}{d x} [12 (x^{1} x^{3})] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 2.3

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

$\frac{d}{d x} [12 x^{1 + 3}] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 2.4

Add $1$ and $3$ .

$\frac{d}{d x} [12 x^{4}] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 2.5

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{4}$ with respect to $x$ is $12 \frac{d}{d x} [x^{4}]$ .

$12 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$12 (4 x^{3}) + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 2.7

Multiply $4$ by $12$ .

$48 x^{3} + \frac{d}{d x} [- \frac{1}{x}] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3

Evaluate $\frac{d}{d x} [- \frac{1}{x}]$ .

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Step 3.1

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 $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

$48 x^{3} - \frac{d}{d x} [\frac{1}{x}] + \frac{1}{x} \frac{d}{d x} [- 1] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3.2

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

$48 x^{3} - \frac{d}{d x} [x^{- 1}] + \frac{1}{x} \frac{d}{d x} [- 1] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3.3

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

$48 x^{3} - - x^{- 2} + \frac{1}{x} \frac{d}{d x} [- 1] + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3.4

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

$48 x^{3} - - x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3.5

Multiply $- 1$ by $- 1$ .

$48 x^{3} + 1 x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3.6

Multiply $x^{- 2}$ by $1$ .

$48 x^{3} + x^{- 2} + \frac{1}{x} \cdot 0 + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3.7

Multiply $\frac{1}{x}$ by $0$ .

$48 x^{3} + x^{- 2} + 0 + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 3.8

Add $x^{- 2}$ and $0$ .

$48 x^{3} + x^{- 2} + \frac{d}{d x} [3 e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 4

Evaluate $\frac{d}{d x} [3 e^{x}]$ .

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Step 4.1

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

$48 x^{3} + x^{- 2} + 3 \frac{d}{d x} [e^{x}] + \frac{d}{d x} [- \sin (x)]$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$48 x^{3} + x^{- 2} + 3 e^{x} + \frac{d}{d x} [- \sin (x)]$

Step 5

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

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Step 5.1

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

$48 x^{3} + x^{- 2} + 3 e^{x} - \frac{d}{d x} [\sin (x)]$

Step 5.2

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

$48 x^{3} + x^{- 2} + 3 e^{x} - \cos (x)$

Step 6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$48 x^{3} + \frac{1}{x^{2}} + 3 e^{x} - \cos (x)$