Calculus Examples

$x {(x - 5)}^{2}$

Step 1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

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

Step 2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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 $x$ .

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

Step 3.1.2

Move $- 5$ to the left of $x$ .

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

Step 3.1.3

Multiply $- 5$ by $- 5$ .

$\frac{d}{d x} [x (x^{2} - 5 x - 5 x + 25)]$

Step 3.2

Subtract $5 x$ from $- 5 x$ .

$\frac{d}{d x} [x (x^{2} - 10 x + 25)]$

Step 4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x^{2} - 10 x + 25$ .

$x \frac{d}{d x} [x^{2} - 10 x + 25] + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x^{2} - 10 x + 25$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]$ .

$x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5.2

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

$x (2 x + \frac{d}{d x} [- 10 x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5.3

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10 x$ with respect to $x$ is $- 10 \frac{d}{d x} [x]$ .

$x (2 x - 10 \frac{d}{d x} [x] + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5.4

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

$x (2 x - 10 \cdot 1 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5.5

Multiply $- 10$ by $1$ .

$x (2 x - 10 + \frac{d}{d x} [25]) + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5.6

Since $25$ is constant with respect to $x$ , the derivative of $25$ with respect to $x$ is $0$ .

$x (2 x - 10 + 0) + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5.7

Add $2 x - 10$ and $0$ .

$x (2 x - 10) + (x^{2} - 10 x + 25) \frac{d}{d x} [x]$

Step 5.8

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

$x (2 x - 10) + (x^{2} - 10 x + 25) \cdot 1$

Step 5.9

Multiply $x^{2} - 10 x + 25$ by $1$ .

$x (2 x - 10) + x^{2} - 10 x + 25$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$x (2 x) + x \cdot - 10 + x^{2} - 10 x + 25$

Step 6.2

Combine terms.

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Step 6.2.1

Raise $x$ to the power of $1$ .

$2 (x^{1} x) + x \cdot - 10 + x^{2} - 10 x + 25$

Step 6.2.2

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{1}) + x \cdot - 10 + x^{2} - 10 x + 25$

Step 6.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1} + x \cdot - 10 + x^{2} - 10 x + 25$

Step 6.2.4

Add $1$ and $1$ .

$2 x^{2} + x \cdot - 10 + x^{2} - 10 x + 25$

Step 6.2.5

Move $- 10$ to the left of $x$ .

$2 x^{2} - 10 \cdot x + x^{2} - 10 x + 25$

Step 6.2.6

Add $2 x^{2}$ and $x^{2}$ .

$3 x^{2} - 10 x - 10 x + 25$

Step 6.2.7

Subtract $10 x$ from $- 10 x$ .

$3 x^{2} - 20 x + 25$