Calculus Examples

$\frac{5 x - 7}{35 x^{3}}$

Step 1

Since $\frac{1}{35}$ is constant with respect to $x$ , the derivative of $\frac{5 x - 7}{35 x^{3}}$ with respect to $x$ is $\frac{1}{35} \frac{d}{d x} [\frac{5 x - 7}{x^{3}}]$ .

$\frac{1}{35} \frac{d}{d x} [\frac{5 x - 7}{x^{3}}]$

Step 2

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 $f (x) = 5 x - 7$ and $g (x) = x^{3}$ .

$\frac{1}{35} \cdot \frac{x^{3} \frac{d}{d x} [5 x - 7] - (5 x - 7) \frac{d}{d x} [x^{3}]}{{(x^{3})}^{2}}$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Multiply the exponents in ${(x^{3})}^{2}$ .

Tap for more steps...

Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{35} \cdot \frac{x^{3} \frac{d}{d x} [5 x - 7] - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{3 \cdot 2}}$

Step 3.1.2

Multiply $3$ by $2$ .

$\frac{1}{35} \cdot \frac{x^{3} \frac{d}{d x} [5 x - 7] - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.2

By the Sum Rule, the derivative of $5 x - 7$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 7]$ .

$\frac{1}{35} \cdot \frac{x^{3} (\frac{d}{d x} [5 x] + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{1}{35} \cdot \frac{x^{3} (5 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.4

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

$\frac{1}{35} \cdot \frac{x^{3} (5 \cdot 1 + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.5

Multiply $5$ by $1$ .

$\frac{1}{35} \cdot \frac{x^{3} (5 + \frac{d}{d x} [- 7]) - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.6

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\frac{1}{35} \cdot \frac{x^{3} (5 + 0) - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.7

Simplify the expression.

Tap for more steps...

Step 3.7.1

Add $5$ and $0$ .

$\frac{1}{35} \cdot \frac{x^{3} \cdot 5 - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.7.2

Move $5$ to the left of $x^{3}$ .

$\frac{1}{35} \cdot \frac{5 \cdot x^{3} - (5 x - 7) \frac{d}{d x} [x^{3}]}{x^{6}}$

Step 3.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{35} \cdot \frac{5 x^{3} - (5 x - 7) (3 x^{2})}{x^{6}}$

Step 3.9

Simplify with factoring out.

Tap for more steps...

Step 3.9.1

Multiply $3$ by $- 1$ .

$\frac{1}{35} \cdot \frac{5 x^{3} - 3 (5 x - 7) x^{2}}{x^{6}}$

Step 3.9.2

Factor $x^{2}$ out of $5 x^{3} - 3 (5 x - 7) x^{2}$ .

Tap for more steps...

Step 3.9.2.1

Factor $x^{2}$ out of $5 x^{3}$ .

$\frac{1}{35} \cdot \frac{x^{2} (5 x) - 3 (5 x - 7) x^{2}}{x^{6}}$

Step 3.9.2.2

Factor $x^{2}$ out of $- 3 (5 x - 7) x^{2}$ .

$\frac{1}{35} \cdot \frac{x^{2} (5 x) + x^{2} (- 3 (5 x - 7))}{x^{6}}$

Step 3.9.2.3

Factor $x^{2}$ out of $x^{2} (5 x) + x^{2} (- 3 (5 x - 7))$ .

$\frac{1}{35} \cdot \frac{x^{2} (5 x - 3 (5 x - 7))}{x^{6}}$

Step 4

Cancel the common factors.

Tap for more steps...

Step 4.1

Factor $x^{2}$ out of $x^{6}$ .

$\frac{1}{35} \cdot \frac{x^{2} (5 x - 3 (5 x - 7))}{x^{2} x^{4}}$

Step 4.2

Cancel the common factor.

$\frac{1}{35} \cdot \frac{x^{2} (5 x - 3 (5 x - 7))}{x^{2} x^{4}}$

Step 4.3

Rewrite the expression.

$\frac{1}{35} \cdot \frac{5 x - 3 (5 x - 7)}{x^{4}}$

Step 5

Multiply $\frac{1}{35}$ by $\frac{5 x - 3 (5 x - 7)}{x^{4}}$ .

$\frac{5 x - 3 (5 x - 7)}{35 x^{4}}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{5 x - 3 (5 x) - 3 \cdot - 7}{35 x^{4}}$

Step 6.2

Simplify the numerator.

Tap for more steps...

Step 6.2.1

Simplify each term.

Tap for more steps...

Step 6.2.1.1

Multiply $5$ by $- 3$ .

$\frac{5 x - 15 x - 3 \cdot - 7}{35 x^{4}}$

Step 6.2.1.2

Multiply $- 3$ by $- 7$ .

$\frac{5 x - 15 x + 21}{35 x^{4}}$

Step 6.2.2

Subtract $15 x$ from $5 x$ .

$\frac{- 10 x + 21}{35 x^{4}}$

Step 6.3

Factor $- 1$ out of $- 10 x$ .

$\frac{- (10 x) + 21}{35 x^{4}}$

Step 6.4

Rewrite $21$ as $- 1 (- 21)$ .

$\frac{- (10 x) - 1 (- 21)}{35 x^{4}}$

Step 6.5

Factor $- 1$ out of $- (10 x) - 1 (- 21)$ .

$\frac{- (10 x - 21)}{35 x^{4}}$

Step 6.6

Rewrite $- (10 x - 21)$ as $- 1 (10 x - 21)$ .

$\frac{- 1 (10 x - 21)}{35 x^{4}}$

Step 6.7

Move the negative in front of the fraction.

$- \frac{10 x - 21}{35 x^{4}}$