Calculus Examples

$(3 x^{3} + 1) {(- 4 x^{2} - 3)}^{4}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 3 x^{3} + 1$ and $g (x) = {(- 4 x^{2} - 3)}^{4}$ .

$(3 x^{3} + 1) \frac{d}{d x} [{(- 4 x^{2} - 3)}^{4}] + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = - 4 x^{2} - 3$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- 4 x^{2} - 3$ .

$(3 x^{3} + 1) (\frac{d}{d u} [u^{4}] \frac{d}{d x} [- 4 x^{2} - 3]) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$(3 x^{3} + 1) (4 u^{3} \frac{d}{d x} [- 4 x^{2} - 3]) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 2.3

Replace all occurrences of $u$ with $- 4 x^{2} - 3$ .

$(3 x^{3} + 1) (4 {(- 4 x^{2} - 3)}^{3} \frac{d}{d x} [- 4 x^{2} - 3]) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3

Differentiate.

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Step 3.1

Move $4$ to the left of $3 x^{3} + 1$ .

$4 \cdot (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} \frac{d}{d x} [- 4 x^{2} - 3] + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.2

By the Sum Rule, the derivative of $- 4 x^{2} - 3$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 3]$ .

$4 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} (\frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 3]) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.3

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$4 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} (- 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3]) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.4

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

$4 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} (- 4 (2 x) + \frac{d}{d x} [- 3]) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.5

Multiply $2$ by $- 4$ .

$4 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} (- 8 x + \frac{d}{d x} [- 3]) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.6

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

$4 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} (- 8 x + 0) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $- 8 x$ and $0$ .

$4 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} (- 8 x) + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.7.2

Multiply $- 8$ by $4$ .

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + {(- 4 x^{2} - 3)}^{4} \frac{d}{d x} [3 x^{3} + 1]$

Step 3.8

By the Sum Rule, the derivative of $3 x^{3} + 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [1]$ .

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + {(- 4 x^{2} - 3)}^{4} (\frac{d}{d x} [3 x^{3}] + \frac{d}{d x} [1])$

Step 3.9

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

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + {(- 4 x^{2} - 3)}^{4} (3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [1])$

Step 3.10

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

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + {(- 4 x^{2} - 3)}^{4} (3 (3 x^{2}) + \frac{d}{d x} [1])$

Step 3.11

Multiply $3$ by $3$ .

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + {(- 4 x^{2} - 3)}^{4} (9 x^{2} + \frac{d}{d x} [1])$

Step 3.12

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

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + {(- 4 x^{2} - 3)}^{4} (9 x^{2} + 0)$

Step 3.13

Simplify the expression.

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Step 3.13.1

Add $9 x^{2}$ and $0$ .

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + {(- 4 x^{2} - 3)}^{4} (9 x^{2})$

Step 3.13.2

Move $9$ to the left of ${(- 4 x^{2} - 3)}^{4}$ .

$- 32 (3 x^{3} + 1) {(- 4 x^{2} - 3)}^{3} x + 9 \cdot {(- 4 x^{2} - 3)}^{4} x^{2}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(- 32 (3 x^{3}) - 32 \cdot 1) {(- 4 x^{2} - 3)}^{3} x + 9 {(- 4 x^{2} - 3)}^{4} x^{2}$

Step 4.2

Multiply $3$ by $- 32$ .

$(- 96 x^{3} - 32 \cdot 1) {(- 4 x^{2} - 3)}^{3} x + 9 {(- 4 x^{2} - 3)}^{4} x^{2}$

Step 4.3

Multiply $- 32$ by $1$ .

$(- 96 x^{3} - 32) {(- 4 x^{2} - 3)}^{3} x + 9 {(- 4 x^{2} - 3)}^{4} x^{2}$

Step 4.4

Factor ${(- 4 x^{2} - 3)}^{3} x$ out of $(- 96 x^{3} - 32) {(- 4 x^{2} - 3)}^{3} x + 9 {(- 4 x^{2} - 3)}^{4} x^{2}$ .

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Step 4.4.1

Factor ${(- 4 x^{2} - 3)}^{3} x$ out of $(- 96 x^{3} - 32) {(- 4 x^{2} - 3)}^{3} x$ .

${(- 4 x^{2} - 3)}^{3} x (- 96 x^{3} - 32) + 9 {(- 4 x^{2} - 3)}^{4} x^{2}$

Step 4.4.2

Factor ${(- 4 x^{2} - 3)}^{3} x$ out of $9 {(- 4 x^{2} - 3)}^{4} x^{2}$ .

${(- 4 x^{2} - 3)}^{3} x (- 96 x^{3} - 32) + {(- 4 x^{2} - 3)}^{3} x (9 (- 4 x^{2} - 3) x)$

Step 4.4.3

Factor ${(- 4 x^{2} - 3)}^{3} x$ out of ${(- 4 x^{2} - 3)}^{3} x (- 96 x^{3} - 32) + {(- 4 x^{2} - 3)}^{3} x (9 (- 4 x^{2} - 3) x)$ .

${(- 4 x^{2} - 3)}^{3} x (- 96 x^{3} - 32 + 9 (- 4 x^{2} - 3) x)$

Step 4.5

Reorder the factors of ${(- 4 x^{2} - 3)}^{3} x (- 96 x^{3} - 32 + 9 (- 4 x^{2} - 3) x)$ .

$x {(- 4 x^{2} - 3)}^{3} (- 96 x^{3} - 32 + 9 (- 4 x^{2} - 3) x)$