Calculus Examples

$f (x) x^{3} - 3 x^{2} - x - 5$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Move $3$ to the left of $f (x)$ .

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

Step 3

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Multiply $2$ by $- 3$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

$3 f (x) x^{2} - 6 x - 1 \cdot 1 + \frac{d}{d x} [- 5]$

Step 4.3

Multiply $- 1$ by $1$ .

$3 f (x) x^{2} - 6 x - 1 + \frac{d}{d x} [- 5]$

Step 5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$3 f (x) x^{2} - 6 x - 1 + 0$

Step 6

Simplify.

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Step 6.1

Add $3 f (x) x^{2} - 6 x - 1$ and $0$ .

$3 f (x) x^{2} - 6 x - 1$

Step 6.2

Reorder terms.

$3 x^{2} f (x) - 6 x - 1$