Calculus Examples

${(3 x^{2} - 7 x - 3)}^{2} \cdot 16$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Move $16$ to the left of ${(3 x^{2} - 7 x - 3)}^{2}$ .

$\frac{d}{d x} [16 \cdot {(3 x^{2} - 7 x - 3)}^{2}]$

Step 1.2

Since $16$ is constant with respect to $x$ , the derivative of $16 {(3 x^{2} - 7 x - 3)}^{2}$ with respect to $x$ is $16 \frac{d}{d x} [{(3 x^{2} - 7 x - 3)}^{2}]$ .

$16 \frac{d}{d x} [{(3 x^{2} - 7 x - 3)}^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = 3 x^{2} - 7 x - 3$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x^{2} - 7 x - 3$ .

$16 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [3 x^{2} - 7 x - 3])$

Step 2.2

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

$16 (2 u \frac{d}{d x} [3 x^{2} - 7 x - 3])$

Step 2.3

Replace all occurrences of $u$ with $3 x^{2} - 7 x - 3$ .

$16 (2 (3 x^{2} - 7 x - 3) \frac{d}{d x} [3 x^{2} - 7 x - 3])$

Step 3

Differentiate.

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Step 3.1

Multiply $2$ by $16$ .

$32 ((3 x^{2} - 7 x - 3) \frac{d}{d x} [3 x^{2} - 7 x - 3])$

Step 3.2

By the Sum Rule, the derivative of $3 x^{2} - 7 x - 3$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 3]$ .

$32 (3 x^{2} - 7 x - 3) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 3])$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$32 (3 x^{2} - 7 x - 3) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 3])$

Step 3.4

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

$32 (3 x^{2} - 7 x - 3) (3 (2 x) + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 3])$

Step 3.5

Multiply $2$ by $3$ .

$32 (3 x^{2} - 7 x - 3) (6 x + \frac{d}{d x} [- 7 x] + \frac{d}{d x} [- 3])$

Step 3.6

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

$32 (3 x^{2} - 7 x - 3) (6 x - 7 \frac{d}{d x} [x] + \frac{d}{d x} [- 3])$

Step 3.7

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

$32 (3 x^{2} - 7 x - 3) (6 x - 7 \cdot 1 + \frac{d}{d x} [- 3])$

Step 3.8

Multiply $- 7$ by $1$ .

$32 (3 x^{2} - 7 x - 3) (6 x - 7 + \frac{d}{d x} [- 3])$

Step 3.9

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$32 (3 x^{2} - 7 x - 3) (6 x - 7 + 0)$

Step 3.10

Add $6 x - 7$ and $0$ .

$32 (3 x^{2} - 7 x - 3) (6 x - 7)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(32 (3 x^{2}) + 32 (- 7 x) + 32 \cdot - 3) (6 x - 7)$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $3$ by $32$ .

$(96 x^{2} + 32 (- 7 x) + 32 \cdot - 3) (6 x - 7)$

Step 4.2.2

Multiply $- 7$ by $32$ .

$(96 x^{2} - 224 x + 32 \cdot - 3) (6 x - 7)$

Step 4.2.3

Multiply $32$ by $- 3$ .

$(96 x^{2} - 224 x - 96) (6 x - 7)$

Step 4.3

Reorder the factors of $(96 x^{2} - 224 x - 96) (6 x - 7)$ .

$(6 x - 7) (96 x^{2} - 224 x - 96)$