Calculus Examples

$f (x) = {(2 x - 1)}^{4} {(3 x - 2)}^{6}$

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 $f (x) = {(2 x - 1)}^{4}$ and $g (x) = {(3 x - 2)}^{6}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = 3 x - 2$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $3 x - 2$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 6$ .

${(2 x - 1)}^{4} (6 {u_{1}}^{5} \frac{d}{d x} [3 x - 2]) + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $3 x - 2$ .

${(2 x - 1)}^{4} (6 {(3 x - 2)}^{5} \frac{d}{d x} [3 x - 2]) + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3

Differentiate.

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Step 3.1

Move $6$ to the left of ${(2 x - 1)}^{4}$ .

$6 \cdot {(2 x - 1)}^{4} {(3 x - 2)}^{5} \frac{d}{d x} [3 x - 2] + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3.2

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

$6 {(2 x - 1)}^{4} {(3 x - 2)}^{5} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2]) + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3.3

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

$6 {(2 x - 1)}^{4} {(3 x - 2)}^{5} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2]) + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3.4

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

$6 {(2 x - 1)}^{4} {(3 x - 2)}^{5} (3 \cdot 1 + \frac{d}{d x} [- 2]) + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3.5

Multiply $3$ by $1$ .

$6 {(2 x - 1)}^{4} {(3 x - 2)}^{5} (3 + \frac{d}{d x} [- 2]) + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3.6

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

$6 {(2 x - 1)}^{4} {(3 x - 2)}^{5} (3 + 0) + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $3$ and $0$ .

$6 {(2 x - 1)}^{4} {(3 x - 2)}^{5} \cdot 3 + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 3.7.2

Multiply $3$ by $6$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + {(3 x - 2)}^{6} \frac{d}{d x} [{(2 x - 1)}^{4}]$

Step 4

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 $g (x) = 2 x - 1$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $2 x - 1$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + {(3 x - 2)}^{6} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [2 x - 1])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + {(3 x - 2)}^{6} (4 {u_{2}}^{3} \frac{d}{d x} [2 x - 1])$

Step 4.3

Replace all occurrences of $u_{2}$ with $2 x - 1$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + {(3 x - 2)}^{6} (4 {(2 x - 1)}^{3} \frac{d}{d x} [2 x - 1])$

Step 5

Differentiate.

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Step 5.1

Move $4$ to the left of ${(3 x - 2)}^{6}$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 4 \cdot {(3 x - 2)}^{6} {(2 x - 1)}^{3} \frac{d}{d x} [2 x - 1]$

Step 5.2

By the Sum Rule, the derivative of $2 x - 1$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1]$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 4 {(3 x - 2)}^{6} {(2 x - 1)}^{3} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 1])$

Step 5.3

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

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 4 {(3 x - 2)}^{6} {(2 x - 1)}^{3} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 5.4

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

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 4 {(3 x - 2)}^{6} {(2 x - 1)}^{3} (2 \cdot 1 + \frac{d}{d x} [- 1])$

Step 5.5

Multiply $2$ by $1$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 4 {(3 x - 2)}^{6} {(2 x - 1)}^{3} (2 + \frac{d}{d x} [- 1])$

Step 5.6

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

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 4 {(3 x - 2)}^{6} {(2 x - 1)}^{3} (2 + 0)$

Step 5.7

Simplify the expression.

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Step 5.7.1

Add $2$ and $0$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 4 {(3 x - 2)}^{6} {(2 x - 1)}^{3} \cdot 2$

Step 5.7.2

Multiply $2$ by $4$ .

$18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 8 {(3 x - 2)}^{6} {(2 x - 1)}^{3}$

Step 6

Simplify.

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Step 6.1

Factor $2 {(2 x - 1)}^{3} {(3 x - 2)}^{5}$ out of $18 {(2 x - 1)}^{4} {(3 x - 2)}^{5} + 8 {(3 x - 2)}^{6} {(2 x - 1)}^{3}$ .

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Step 6.1.1

Factor $2 {(2 x - 1)}^{3} {(3 x - 2)}^{5}$ out of $18 {(2 x - 1)}^{4} {(3 x - 2)}^{5}$ .

$2 {(2 x - 1)}^{3} {(3 x - 2)}^{5} (9 (2 x - 1)) + 8 {(3 x - 2)}^{6} {(2 x - 1)}^{3}$

Step 6.1.2

Factor $2 {(2 x - 1)}^{3} {(3 x - 2)}^{5}$ out of $8 {(3 x - 2)}^{6} {(2 x - 1)}^{3}$ .

$2 {(2 x - 1)}^{3} {(3 x - 2)}^{5} (9 (2 x - 1)) + 2 {(2 x - 1)}^{3} {(3 x - 2)}^{5} (4 (3 x - 2))$

Step 6.1.3

Factor $2 {(2 x - 1)}^{3} {(3 x - 2)}^{5}$ out of $2 {(2 x - 1)}^{3} {(3 x - 2)}^{5} (9 (2 x - 1)) + 2 {(2 x - 1)}^{3} {(3 x - 2)}^{5} (4 (3 x - 2))$ .

$2 {(2 x - 1)}^{3} {(3 x - 2)}^{5} (9 (2 x - 1) + 4 (3 x - 2))$

Step 6.2

Reorder the factors of $2 {(2 x - 1)}^{3} {(3 x - 2)}^{5} (9 (2 x - 1) + 4 (3 x - 2))$ .

$2 {(3 x - 2)}^{5} {(2 x - 1)}^{3} (9 (2 x - 1) + 4 (3 x - 2))$