Calculus Examples

$7 x^{2} {(6 x + 3)}^{4}$

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 $f (x) = 7 x^{2}$ and $g (x) = {(6 x + 3)}^{4}$ .

$7 x^{2} \frac{d}{d x} [{(6 x + 3)}^{4}] + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 6 x + 3$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $6 x + 3$ .

$7 x^{2} (\frac{d}{d u} [u^{4}] \frac{d}{d x} [6 x + 3]) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 2.2

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

$7 x^{2} (4 u^{3} \frac{d}{d x} [6 x + 3]) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $6 x + 3$ .

$7 x^{2} (4 {(6 x + 3)}^{3} \frac{d}{d x} [6 x + 3]) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $6 x + 3$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [3]$ .

$7 x^{2} (4 {(6 x + 3)}^{3} (\frac{d}{d x} [6 x] + \frac{d}{d x} [3])) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$7 x^{2} (4 {(6 x + 3)}^{3} (6 \frac{d}{d x} [x] + \frac{d}{d x} [3])) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.3

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

$7 x^{2} (4 {(6 x + 3)}^{3} (6 \cdot 1 + \frac{d}{d x} [3])) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.4

Multiply $6$ by $1$ .

$7 x^{2} (4 {(6 x + 3)}^{3} (6 + \frac{d}{d x} [3])) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.5

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

$7 x^{2} (4 {(6 x + 3)}^{3} (6 + 0)) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $6$ and $0$ .

$7 x^{2} (4 {(6 x + 3)}^{3} \cdot 6) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.6.2

Multiply $6$ by $4$ .

$7 x^{2} (24 {(6 x + 3)}^{3}) + {(6 x + 3)}^{4} \frac{d}{d x} [7 x^{2}]$

Step 3.7

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

$7 x^{2} (24 {(6 x + 3)}^{3}) + {(6 x + 3)}^{4} (7 \frac{d}{d x} [x^{2}])$

Step 3.8

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

$7 x^{2} (24 {(6 x + 3)}^{3}) + {(6 x + 3)}^{4} (7 (2 x))$

Step 3.9

Multiply $2$ by $7$ .

$7 x^{2} (24 {(6 x + 3)}^{3}) + {(6 x + 3)}^{4} (14 x)$

Step 4

Simplify.

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Step 4.1

Multiply $24$ by $7$ .

$168 x^{2} {(6 x + 3)}^{3} + {(6 x + 3)}^{4} (14 x)$

Step 4.2

Move $14$ to the left of ${(6 x + 3)}^{4}$ .

$168 x^{2} {(6 x + 3)}^{3} + 14 \cdot {(6 x + 3)}^{4} x$

Step 4.3

Factor $14 x {(6 x + 3)}^{3}$ out of $168 x^{2} {(6 x + 3)}^{3} + 14 {(6 x + 3)}^{4} x$ .

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Step 4.3.1

Factor $14 x {(6 x + 3)}^{3}$ out of $168 x^{2} {(6 x + 3)}^{3}$ .

$14 x {(6 x + 3)}^{3} (12 x) + 14 {(6 x + 3)}^{4} x$

Step 4.3.2

Factor $14 x {(6 x + 3)}^{3}$ out of $14 {(6 x + 3)}^{4} x$ .

$14 x {(6 x + 3)}^{3} (12 x) + 14 x {(6 x + 3)}^{3} (6 x + 3)$

Step 4.3.3

Factor $14 x {(6 x + 3)}^{3}$ out of $14 x {(6 x + 3)}^{3} (12 x) + 14 x {(6 x + 3)}^{3} (6 x + 3)$ .

$14 x {(6 x + 3)}^{3} (12 x + 6 x + 3)$

Step 4.4

Add $12 x$ and $6 x$ .

$14 x {(6 x + 3)}^{3} (18 x + 3)$