Calculus Examples

$840 - 12 q^{7} q^{2} + 2 q$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $840 - 12 q^{7} q^{2} + 2 q$ with respect to $q$ is $\frac{d}{d q} [840] + \frac{d}{d q} [- 12 q^{7} q^{2}] + \frac{d}{d q} [2 q]$ .

$\frac{d}{d q} [840] + \frac{d}{d q} [- 12 q^{7} q^{2}] + \frac{d}{d q} [2 q]$

Step 1.2

Since $840$ is constant with respect to $q$ , the derivative of $840$ with respect to $q$ is $0$ .

$0 + \frac{d}{d q} [- 12 q^{7} q^{2}] + \frac{d}{d q} [2 q]$

Step 2

Evaluate $\frac{d}{d q} [- 12 q^{7} q^{2}]$ .

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Step 2.1

Multiply $q^{7}$ by $q^{2}$ by adding the exponents.

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Step 2.1.1

Move $q^{2}$ .

$0 + \frac{d}{d q} [- 12 (q^{2} q^{7})] + \frac{d}{d q} [2 q]$

Step 2.1.2

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

$0 + \frac{d}{d q} [- 12 q^{2 + 7}] + \frac{d}{d q} [2 q]$

Step 2.1.3

Add $2$ and $7$ .

$0 + \frac{d}{d q} [- 12 q^{9}] + \frac{d}{d q} [2 q]$

Step 2.2

Since $- 12$ is constant with respect to $q$ , the derivative of $- 12 q^{9}$ with respect to $q$ is $- 12 \frac{d}{d q} [q^{9}]$ .

$0 - 12 \frac{d}{d q} [q^{9}] + \frac{d}{d q} [2 q]$

Step 2.3

Differentiate using the Power Rule which states that $\frac{d}{d q} [q^{n}]$ is $n q^{n - 1}$ where $n = 9$ .

$0 - 12 (9 q^{8}) + \frac{d}{d q} [2 q]$

Step 2.4

Multiply $9$ by $- 12$ .

$0 - 108 q^{8} + \frac{d}{d q} [2 q]$

Step 3

Evaluate $\frac{d}{d q} [2 q]$ .

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Step 3.1

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

$0 - 108 q^{8} + 2 \frac{d}{d q} [q]$

Step 3.2

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

$0 - 108 q^{8} + 2 \cdot 1$

Step 3.3

Multiply $2$ by $1$ .

$0 - 108 q^{8} + 2$

Step 4

Subtract $108 q^{8}$ from $0$ .

$- 108 q^{8} + 2$