Calculus Examples

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

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 $f (x) = {(3 x - 1)}^{4}$ and $g (x) = {(- 2 x + 9)}^{5}$ .

${(3 x - 1)}^{4} \frac{d}{d x} [{(- 2 x + 9)}^{5}] + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = - 2 x + 9$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{1}$ as $- 2 x + 9$ .

${(3 x - 1)}^{4} (\frac{d}{d u_{1}} [{u_{1}}^{5}] \frac{d}{d x} [- 2 x + 9]) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

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 $n = 5$ .

${(3 x - 1)}^{4} (5 {u_{1}}^{4} \frac{d}{d x} [- 2 x + 9]) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $- 2 x + 9$ .

${(3 x - 1)}^{4} (5 {(- 2 x + 9)}^{4} \frac{d}{d x} [- 2 x + 9]) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

Move $5$ to the left of ${(3 x - 1)}^{4}$ .

$5 \cdot {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} \frac{d}{d x} [- 2 x + 9] + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3.2

By the Sum Rule, the derivative of $- 2 x + 9$ with respect to $x$ is $\frac{d}{d x} [- 2 x] + \frac{d}{d x} [9]$ .

$5 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} (\frac{d}{d x} [- 2 x] + \frac{d}{d x} [9]) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3.3

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

$5 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} (- 2 \frac{d}{d x} [x] + \frac{d}{d x} [9]) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3.4

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

$5 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} (- 2 \cdot 1 + \frac{d}{d x} [9]) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3.5

Multiply $- 2$ by $1$ .

$5 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} (- 2 + \frac{d}{d x} [9]) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3.6

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$5 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} (- 2 + 0) + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3.7

Simplify the expression.

Tap for more steps...

Step 3.7.1

Add $- 2$ and $0$ .

$5 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} \cdot - 2 + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

Step 3.7.2

Multiply $- 2$ by $5$ .

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + {(- 2 x + 9)}^{5} \frac{d}{d x} [{(3 x - 1)}^{4}]$

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 $g (x) = 3 x - 1$ .

Tap for more steps...

Step 4.1

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

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + {(- 2 x + 9)}^{5} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [3 x - 1])$

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 $n = 4$ .

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + {(- 2 x + 9)}^{5} (4 {u_{2}}^{3} \frac{d}{d x} [3 x - 1])$

Step 4.3

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

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + {(- 2 x + 9)}^{5} (4 {(3 x - 1)}^{3} \frac{d}{d x} [3 x - 1])$

Step 5

Differentiate.

Tap for more steps...

Step 5.1

Move $4$ to the left of ${(- 2 x + 9)}^{5}$ .

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 4 \cdot {(- 2 x + 9)}^{5} {(3 x - 1)}^{3} \frac{d}{d x} [3 x - 1]$

Step 5.2

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

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 4 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 1])$

Step 5.3

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

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 4 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 1])$

Step 5.4

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

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 4 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3} (3 \cdot 1 + \frac{d}{d x} [- 1])$

Step 5.5

Multiply $3$ by $1$ .

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 4 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3} (3 + \frac{d}{d x} [- 1])$

Step 5.6

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

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 4 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3} (3 + 0)$

Step 5.7

Simplify the expression.

Tap for more steps...

Step 5.7.1

Add $3$ and $0$ .

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 4 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3} \cdot 3$

Step 5.7.2

Multiply $3$ by $4$ .

$- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 12 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3}$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Factor $2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4}$ out of $- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4} + 12 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3}$ .

Tap for more steps...

Step 6.1.1

Factor $2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4}$ out of $- 10 {(3 x - 1)}^{4} {(- 2 x + 9)}^{4}$ .

$2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4} (- 5 (3 x - 1)) + 12 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3}$

Step 6.1.2

Factor $2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4}$ out of $12 {(- 2 x + 9)}^{5} {(3 x - 1)}^{3}$ .

$2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4} (- 5 (3 x - 1)) + 2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4} (6 (- 2 x + 9))$

Step 6.1.3

Factor $2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4}$ out of $2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4} (- 5 (3 x - 1)) + 2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4} (6 (- 2 x + 9))$ .

$2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4} (- 5 (3 x - 1) + 6 (- 2 x + 9))$

Step 6.2

Reorder the factors of $2 {(3 x - 1)}^{3} {(- 2 x + 9)}^{4} (- 5 (3 x - 1) + 6 (- 2 x + 9))$ .

$2 {(- 2 x + 9)}^{4} {(3 x - 1)}^{3} (- 5 (3 x - 1) + 6 (- 2 x + 9))$