Calculus Examples

$f (x) = 5 {(x - 2)}^{2} + 3$

Step 1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

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

Step 2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ .

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

Step 3.1.2

Move $- 2$ to the left of $x$ .

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

Step 3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 3.2

Subtract $2 x$ from $- 2 x$ .

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

Step 4

By the Sum Rule, the derivative of $5 (x^{2} - 4 x + 4) + 3$ with respect to $x$ is $\frac{d}{d x} [5 (x^{2} - 4 x + 4)] + \frac{d}{d x} [3]$ .

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

Step 5

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

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Step 5.1

Since $5$ is constant with respect to $x$ , the derivative of $5 (x^{2} - 4 x + 4)$ with respect to $x$ is $5 \frac{d}{d x} [x^{2} - 4 x + 4]$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

$5 (2 x - 4 \cdot 1 + \frac{d}{d x} [4]) + \frac{d}{d x} [3]$

Step 5.6

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

$5 (2 x - 4 \cdot 1 + 0) + \frac{d}{d x} [3]$

Step 5.7

Multiply $- 4$ by $1$ .

$5 (2 x - 4 + 0) + \frac{d}{d x} [3]$

Step 5.8

Add $2 x - 4$ and $0$ .

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

Step 6

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$5 (2 x - 4) + 0$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$5 (2 x) + 5 \cdot - 4 + 0$

Step 7.2

Combine terms.

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Step 7.2.1

Multiply $2$ by $5$ .

$10 x + 5 \cdot - 4 + 0$

Step 7.2.2

Multiply $5$ by $- 4$ .

$10 x - 20 + 0$

Step 7.2.3

Add $10 x - 20$ and $0$ .

$10 x - 20$