Calculus Examples

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

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x + 2$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 7$ .

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

Step 2.3

Replace all occurrences of $u$ with $3 x + 2$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $3$ by $1$ .

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

Step 3.5

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

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

Step 3.6.2

Multiply $3$ by $7$ .

$x (21 {(3 x + 2)}^{6}) + {(3 x + 2)}^{7} \frac{d}{d x} [x]$

Step 3.7

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

$x (21 {(3 x + 2)}^{6}) + {(3 x + 2)}^{7} \cdot 1$

Step 3.8

Multiply ${(3 x + 2)}^{7}$ by $1$ .

$x (21 {(3 x + 2)}^{6}) + {(3 x + 2)}^{7}$

Step 4

Simplify.

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Step 4.1

Factor ${(3 x + 2)}^{6}$ out of $x (21 {(3 x + 2)}^{6}) + {(3 x + 2)}^{7}$ .

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Step 4.1.1

Factor ${(3 x + 2)}^{6}$ out of $x (21 {(3 x + 2)}^{6})$ .

${(3 x + 2)}^{6} (x \cdot (21)) + {(3 x + 2)}^{7}$

Step 4.1.2

Factor ${(3 x + 2)}^{6}$ out of ${(3 x + 2)}^{7}$ .

${(3 x + 2)}^{6} (x \cdot (21)) + {(3 x + 2)}^{6} (3 x + 2)$

Step 4.1.3

Factor ${(3 x + 2)}^{6}$ out of ${(3 x + 2)}^{6} (x \cdot (21)) + {(3 x + 2)}^{6} (3 x + 2)$ .

${(3 x + 2)}^{6} (x \cdot (21) + 3 x + 2)$

Step 4.2

Combine terms.

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Step 4.2.1

Move $21$ to the left of $x$ .

${(3 x + 2)}^{6} (21 \cdot x + 3 x + 2)$

Step 4.2.2

Add $21 x$ and $3 x$ .

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