Calculus Examples

$f (x) = x^{4} - 5 x^{2} + 4$ , $x = 1$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{4} - 5 x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [4]$

Step 1.2

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

$4 x^{3} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [4]$

Step 2

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 2.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$4 x^{3} - 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 2.2

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

$4 x^{3} - 5 (2 x) + \frac{d}{d x} [4]$

Step 2.3

Multiply $2$ by $- 5$ .

$4 x^{3} - 10 x + \frac{d}{d x} [4]$

Step 3

Differentiate using the Constant Rule.

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$4 x^{3} - 10 x + 0$

Step 3.2

Add $4 x^{3} - 10 x$ and $0$ .

$4 x^{3} - 10 x$

Step 4

Evaluate the derivative at $x = 1$ .

$4 {(1)}^{3} - 10 \cdot 1$

Step 5

Simplify.

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Step 5.1

Simplify each term.

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Step 5.1.1

One to any power is one.

$4 \cdot 1 - 10 \cdot 1$

Step 5.1.2

Multiply $4$ by $1$ .

$4 - 10 \cdot 1$

Step 5.1.3

Multiply $- 10$ by $1$ .

$4 - 10$

Step 5.2

Subtract $10$ from $4$ .

$- 6$