Calculus Examples

$y = \frac{x^{2} - 25}{x - 5}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} - 25$ and $g (x) = x - 5$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$\frac{(x - 5) (2 x + 0) - (x^{2} - 25) \frac{d}{d x} [x - 5]}{{(x - 5)}^{2}}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

Move $2$ to the left of $x - 5$ .

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

Step 2.5

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

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

Step 2.6

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

$\frac{2 (x - 5) x - (x^{2} - 25) (1 + \frac{d}{d x} [- 5])}{{(x - 5)}^{2}}$

Step 2.7

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$\frac{2 (x - 5) x - (x^{2} - 25) (1 + 0)}{{(x - 5)}^{2}}$

Step 2.8

Simplify the expression.

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Step 2.8.1

Add $1$ and $0$ .

$\frac{2 (x - 5) x - (x^{2} - 25) \cdot 1}{{(x - 5)}^{2}}$

Step 2.8.2

Multiply $- 1$ by $1$ .

$\frac{2 (x - 5) x - (x^{2} - 25)}{{(x - 5)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(2 x + 2 \cdot - 5) x - (x^{2} - 25)}{{(x - 5)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{2 x \cdot x + 2 \cdot - 5 x - (x^{2} - 25)}{{(x - 5)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{2 x \cdot x + 2 \cdot - 5 x - x^{2} - - 25}{{(x - 5)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.4.1.1.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 \cdot - 5 x - x^{2} - - 25}{{(x - 5)}^{2}}$

Step 3.4.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 \cdot - 5 x - x^{2} - - 25}{{(x - 5)}^{2}}$

Step 3.4.1.2

Multiply $2$ by $- 5$ .

$\frac{2 x^{2} - 10 x - x^{2} - - 25}{{(x - 5)}^{2}}$

Step 3.4.1.3

Multiply $- 1$ by $- 25$ .

$\frac{2 x^{2} - 10 x - x^{2} + 25}{{(x - 5)}^{2}}$

Step 3.4.2

Subtract $x^{2}$ from $2 x^{2}$ .

$\frac{x^{2} - 10 x + 25}{{(x - 5)}^{2}}$

Step 3.5

Factor using the perfect square rule.

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Step 3.5.1

Rewrite $25$ as $5^{2}$ .

$\frac{x^{2} - 10 x + 5^{2}}{{(x - 5)}^{2}}$

Step 3.5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 3.5.3

Rewrite the polynomial.

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

Step 3.5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

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

Step 3.6

Cancel the common factor of ${(x - 5)}^{2}$ .

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

$1$