Calculus Examples

$y = 24 x - 4 x^{4}$ , $x = 1$

Step 1

By the Sum Rule, the derivative of $24 x - 4 x^{4}$ with respect to $x$ is $\frac{d}{d x} [24 x] + \frac{d}{d x} [- 4 x^{4}]$ .

$\frac{d}{d x} [24 x] + \frac{d}{d x} [- 4 x^{4}]$

Step 2

Evaluate $\frac{d}{d x} [24 x]$ .

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Step 2.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$24 \frac{d}{d x} [x] + \frac{d}{d x} [- 4 x^{4}]$

Step 2.2

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

$24 \cdot 1 + \frac{d}{d x} [- 4 x^{4}]$

Step 2.3

Multiply $24$ by $1$ .

$24 + \frac{d}{d x} [- 4 x^{4}]$

Step 3

Evaluate $\frac{d}{d x} [- 4 x^{4}]$ .

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Step 3.1

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

$24 - 4 \frac{d}{d x} [x^{4}]$

Step 3.2

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

$24 - 4 (4 x^{3})$

Step 3.3

Multiply $4$ by $- 4$ .

$24 - 16 x^{3}$

Step 4

Reorder terms.

$- 16 x^{3} + 24$

Step 5

Evaluate the derivative at $x = 1$ .

$- 16 {(1)}^{3} + 24$

Step 6

Simplify.

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Step 6.1

Simplify each term.

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Step 6.1.1

One to any power is one.

$- 16 \cdot 1 + 24$

Step 6.1.2

Multiply $- 16$ by $1$ .

$- 16 + 24$

Step 6.2

Add $- 16$ and $24$ .

$8$