Calculus Examples

$y = x^{3} + x^{2} - 8 x + 5$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{3} + x^{2} - 8 x + 5$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 8 x] + \frac{d}{d x} [5]$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Evaluate $\frac{d}{d x} [- 8 x]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $- 8$ by $1$ .

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

Step 3

Differentiate using the Constant Rule.

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Step 3.1

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

$3 x^{2} + 2 x - 8 + 0$

Step 3.2

Add $3 x^{2} + 2 x - 8$ and $0$ .

$3 x^{2} + 2 x - 8$