Calculus Examples

$\frac{49 x^{2} - 64}{x}$

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 $f (x) = 49 x^{2} - 64$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [49 x^{2} - 64] - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $49 x^{2} - 64$ with respect to $x$ is $\frac{d}{d x} [49 x^{2}] + \frac{d}{d x} [- 64]$ .

$\frac{x (\frac{d}{d x} [49 x^{2}] + \frac{d}{d x} [- 64]) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.2

Since $49$ is constant with respect to $x$ , the derivative of $49 x^{2}$ with respect to $x$ is $49 \frac{d}{d x} [x^{2}]$ .

$\frac{x (49 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 64]) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.3

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

$\frac{x (49 (2 x) + \frac{d}{d x} [- 64]) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.4

Multiply $2$ by $49$ .

$\frac{x (98 x + \frac{d}{d x} [- 64]) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.5

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64$ with respect to $x$ is $0$ .

$\frac{x (98 x + 0) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.6

Add $98 x$ and $0$ .

$\frac{x (98 x) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 3

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

$\frac{98 (x^{1} x) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 4

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

$\frac{98 (x^{1} x^{1}) - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 5

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

$\frac{98 x^{1 + 1} - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 6

Add $1$ and $1$ .

$\frac{98 x^{2} - (49 x^{2} - 64) \frac{d}{d x} [x]}{x^{2}}$

Step 7

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

$\frac{98 x^{2} - (49 x^{2} - 64) \cdot 1}{x^{2}}$

Step 8

Multiply $- 1$ by $1$ .

$\frac{98 x^{2} - (49 x^{2} - 64)}{x^{2}}$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Apply the distributive property.

$\frac{98 x^{2} - (49 x^{2}) - - 64}{x^{2}}$

Step 9.2

Simplify the numerator.

Tap for more steps...

Step 9.2.1

Simplify each term.

Tap for more steps...

Step 9.2.1.1

Multiply $49$ by $- 1$ .

$\frac{98 x^{2} - 49 x^{2} - - 64}{x^{2}}$

Step 9.2.1.2

Multiply $- 1$ by $- 64$ .

$\frac{98 x^{2} - 49 x^{2} + 64}{x^{2}}$

Step 9.2.2

Subtract $49 x^{2}$ from $98 x^{2}$ .

$\frac{49 x^{2} + 64}{x^{2}}$