Precalculus Examples

$f (x) = - 3 x^{10} - x$

Step 1

By the Sum Rule, the derivative of $- 3 x^{10} - x$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{10}] + \frac{d}{d x} [- x]$ .

$\frac{d}{d x} [- 3 x^{10}] + \frac{d}{d x} [- x]$

Step 2

Evaluate $\frac{d}{d x} [- 3 x^{10}]$ .

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Step 2.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{10}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{10}]$ .

$- 3 \frac{d}{d x} [x^{10}] + \frac{d}{d x} [- x]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 10$ .

$- 3 (10 x^{9}) + \frac{d}{d x} [- x]$

Step 2.3

Multiply $10$ by $- 3$ .

$- 30 x^{9} + \frac{d}{d x} [- x]$

Step 3

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

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Step 3.1

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

$- 30 x^{9} - \frac{d}{d x} [x]$

Step 3.2

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

$- 30 x^{9} - 1 \cdot 1$

Step 3.3

Multiply $- 1$ by $1$ .

$- 30 x^{9} - 1$