Calculus Examples

$y = 34 x - 2 x^{4}$ , $x = 1$

Step 1

By the Sum Rule, the derivative of $34 x - 2 x^{4}$ with respect to $x$ is $\frac{d}{d x} [34 x] + \frac{d}{d x} [- 2 x^{4}]$ .

$\frac{d}{d x} [34 x] + \frac{d}{d x} [- 2 x^{4}]$

Step 2

Evaluate $\frac{d}{d x} [34 x]$ .

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Step 2.1

Since $34$ is constant with respect to $x$ , the derivative of $34 x$ with respect to $x$ is $34 \frac{d}{d x} [x]$ .

$34 \frac{d}{d x} [x] + \frac{d}{d x} [- 2 x^{4}]$

Step 2.2

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

$34 \cdot 1 + \frac{d}{d x} [- 2 x^{4}]$

Step 2.3

Multiply $34$ by $1$ .

$34 + \frac{d}{d x} [- 2 x^{4}]$

Step 3

Evaluate $\frac{d}{d x} [- 2 x^{4}]$ .

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{4}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{4}]$ .

$34 - 2 \frac{d}{d x} [x^{4}]$

Step 3.2

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

$34 - 2 (4 x^{3})$

Step 3.3

Multiply $4$ by $- 2$ .

$34 - 8 x^{3}$

Step 4

Reorder terms.

$- 8 x^{3} + 34$

Step 5

Evaluate the derivative at $x = 1$ .

$- 8 {(1)}^{3} + 34$

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.

$- 8 \cdot 1 + 34$

Step 6.1.2

Multiply $- 8$ by $1$ .

$- 8 + 34$

Step 6.2

Add $- 8$ and $34$ .

$26$