Calculus Examples

$y = \frac{x^{2} - a^{2}}{x - a}$

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 $f (x) = x^{2} - a^{2}$ and $g (x) = x - a$ .

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Since $- a^{2}$ is constant with respect to $x$ , the derivative of $- a^{2}$ with respect to $x$ is $0$ .

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

Step 2.4

Simplify the expression.

Tap for more steps...

Step 2.4.1

Add $2 x$ and $0$ .

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

Step 2.4.2

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Simplify the expression.

Tap for more steps...

Step 2.8.1

Add $1$ and $0$ .

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

Step 2.8.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Simplify the numerator.

Tap for more steps...

Step 3.4.1

Simplify each term.

Tap for more steps...

Step 3.4.1.1

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

Tap for more steps...

Step 3.4.1.1.1

Move $x$ .

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

Step 3.4.1.1.2

Multiply $x$ by $x$ .

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

Step 3.4.1.2

Multiply $- 1$ by $2$ .

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

Step 3.4.1.3

Multiply $- - a^{2}$ .

Tap for more steps...

Step 3.4.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{2 x^{2} - 2 a x - x^{2} + 1 a^{2}}{{(x - a)}^{2}}$

Step 3.4.1.3.2

Multiply $a^{2}$ by $1$ .

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

Step 3.4.2

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

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

Step 3.5

Reorder terms.

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

Step 3.6

Factor using the perfect square rule.

Tap for more steps...

Step 3.6.1

Rearrange terms.

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

Step 3.6.2

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

$2 a x = 2 \cdot a \cdot x$

Step 3.6.3

Rewrite the polynomial.

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

Step 3.6.4

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

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

Step 3.7

Cancel the common factor of ${(a - x)}^{2}$ and ${(- a + x)}^{2}$ .

Tap for more steps...

Step 3.7.1

Factor $- 1$ out of $a$ .

$\frac{{(- 1 (- a) - x)}^{2}}{{(- a + x)}^{2}}$

Step 3.7.2

Factor $- 1$ out of $- x$ .

$\frac{{(- 1 (- a) - (x))}^{2}}{{(- a + x)}^{2}}$

Step 3.7.3

Factor $- 1$ out of $- 1 (- a) - (x)$ .

$\frac{{(- 1 (- a + x))}^{2}}{{(- a + x)}^{2}}$

Step 3.7.4

Apply the product rule to $- 1 (- a + x)$ .

$\frac{{(- 1)}^{2} {(- a + x)}^{2}}{{(- a + x)}^{2}}$

Step 3.7.5

Raise $- 1$ to the power of $2$ .

$\frac{1 {(- a + x)}^{2}}{{(- a + x)}^{2}}$

Step 3.7.6

Multiply ${(- a + x)}^{2}$ by $1$ .

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

Step 3.7.7

Cancel the common factor.

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

Step 3.7.8

Rewrite the expression.

$1$