Calculus Examples

$\frac{q \sin (q)}{q^{2} - 1}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d q} [\frac{f (q)}{g (q)}]$ is $\frac{g (q) \frac{d}{d q} [f (q)] - f (q) \frac{d}{d q} [g (q)]}{{g (q)}^{2}}$ where $f (q) = q \sin (q)$ and $g (q) = q^{2} - 1$ .

$\frac{(q^{2} - 1) \frac{d}{d q} [q \sin (q)] - q \sin (q) \frac{d}{d q} [q^{2} - 1]}{{(q^{2} - 1)}^{2}}$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d q} [f (q) g (q)]$ is $f (q) \frac{d}{d q} [g (q)] + g (q) \frac{d}{d q} [f (q)]$ where $f (q) = q$ and $g (q) = \sin (q)$ .

$\frac{(q^{2} - 1) (q \frac{d}{d q} [\sin (q)] + \sin (q) \frac{d}{d q} [q]) - q \sin (q) \frac{d}{d q} [q^{2} - 1]}{{(q^{2} - 1)}^{2}}$

Step 3

The derivative of $\sin (q)$ with respect to $q$ is $\cos (q)$ .

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q) \frac{d}{d q} [q]) - q \sin (q) \frac{d}{d q} [q^{2} - 1]}{{(q^{2} - 1)}^{2}}$

Step 4

Differentiate.

Tap for more steps...

Step 4.1

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

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q) \cdot 1) - q \sin (q) \frac{d}{d q} [q^{2} - 1]}{{(q^{2} - 1)}^{2}}$

Step 4.2

Multiply $\sin (q)$ by $1$ .

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - q \sin (q) \frac{d}{d q} [q^{2} - 1]}{{(q^{2} - 1)}^{2}}$

Step 4.3

By the Sum Rule, the derivative of $q^{2} - 1$ with respect to $q$ is $\frac{d}{d q} [q^{2}] + \frac{d}{d q} [- 1]$ .

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - q \sin (q) (\frac{d}{d q} [q^{2}] + \frac{d}{d q} [- 1])}{{(q^{2} - 1)}^{2}}$

Step 4.4

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

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - q \sin (q) (2 q + \frac{d}{d q} [- 1])}{{(q^{2} - 1)}^{2}}$

Step 4.5

Since $- 1$ is constant with respect to $q$ , the derivative of $- 1$ with respect to $q$ is $0$ .

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - q \sin (q) (2 q + 0)}{{(q^{2} - 1)}^{2}}$

Step 4.6

Simplify the expression.

Tap for more steps...

Step 4.6.1

Add $2 q$ and $0$ .

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - q \sin (q) (2 q)}{{(q^{2} - 1)}^{2}}$

Step 4.6.2

Multiply $2$ by $- 1$ .

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - 2 q \sin (q) q}{{(q^{2} - 1)}^{2}}$

Step 5

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

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - 2 (q^{1} q) \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 6

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

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - 2 (q^{1} q^{1}) \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 7

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

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - 2 q^{1 + 1} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 8

Add $1$ and $1$ .

$\frac{(q^{2} - 1) (q \cos (q) + \sin (q)) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9

Simplify the numerator.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Expand $(q^{2} - 1) (q \cos (q) + \sin (q))$ using the FOIL Method.

Tap for more steps...

Step 9.1.1.1

Apply the distributive property.

$\frac{q^{2} (q \cos (q) + \sin (q)) - 1 (q \cos (q) + \sin (q)) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.1.2

Apply the distributive property.

$\frac{q^{2} (q \cos (q)) + q^{2} \sin (q) - 1 (q \cos (q) + \sin (q)) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.1.3

Apply the distributive property.

$\frac{q^{2} (q \cos (q)) + q^{2} \sin (q) - 1 (q \cos (q)) - 1 \sin (q) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.2

Simplify each term.

Tap for more steps...

Step 9.1.2.1

Multiply $q^{2}$ by $q$ by adding the exponents.

Tap for more steps...

Step 9.1.2.1.1

Move $q$ .

$\frac{q \cdot q^{2} \cos (q) + q^{2} \sin (q) - 1 (q \cos (q)) - 1 \sin (q) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.2.1.2

Multiply $q$ by $q^{2}$ .

Tap for more steps...

Step 9.1.2.1.2.1

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

$\frac{q^{1} q^{2} \cos (q) + q^{2} \sin (q) - 1 (q \cos (q)) - 1 \sin (q) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.2.1.2.2

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

$\frac{q^{1 + 2} \cos (q) + q^{2} \sin (q) - 1 (q \cos (q)) - 1 \sin (q) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.2.1.3

Add $1$ and $2$ .

$\frac{q^{3} \cos (q) + q^{2} \sin (q) - 1 (q \cos (q)) - 1 \sin (q) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.2.2

Rewrite $- 1 (q \cos (q))$ as $- (q \cos (q))$ .

$\frac{q^{3} \cos (q) + q^{2} \sin (q) - (q \cos (q)) - 1 \sin (q) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.1.2.3

Rewrite $- 1 \sin (q)$ as $- \sin (q)$ .

$\frac{q^{3} \cos (q) + q^{2} \sin (q) - q \cos (q) - \sin (q) - 2 q^{2} \sin (q)}{{(q^{2} - 1)}^{2}}$

Step 9.2

Subtract $2 q^{2} \sin (q)$ from $q^{2} \sin (q)$ .

$\frac{q^{3} \cos (q) - q^{2} \sin (q) - q \cos (q) - \sin (q)}{{(q^{2} - 1)}^{2}}$