Calculus Examples

$f (x) = {(x + 5)}^{7} {(4 x - 1)}^{10}$

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 $f (x) = {(x + 5)}^{7}$ and $g (x) = {(4 x - 1)}^{10}$ .

${(x + 5)}^{7} \frac{d}{d x} [{(4 x - 1)}^{10}] + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{10}$ and $g (x) = 4 x - 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $4 x - 1$ .

${(x + 5)}^{7} (\frac{d}{d u_{1}} [{u_{1}}^{10}] \frac{d}{d x} [4 x - 1]) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

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 $n = 10$ .

${(x + 5)}^{7} (10 {u_{1}}^{9} \frac{d}{d x} [4 x - 1]) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $4 x - 1$ .

${(x + 5)}^{7} (10 {(4 x - 1)}^{9} \frac{d}{d x} [4 x - 1]) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3

Differentiate.

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Step 3.1

Move $10$ to the left of ${(x + 5)}^{7}$ .

$10 \cdot {(x + 5)}^{7} {(4 x - 1)}^{9} \frac{d}{d x} [4 x - 1] + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3.2

By the Sum Rule, the derivative of $4 x - 1$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]$ .

$10 {(x + 5)}^{7} {(4 x - 1)}^{9} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 1]) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3.3

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

$10 {(x + 5)}^{7} {(4 x - 1)}^{9} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 1]) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3.4

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

$10 {(x + 5)}^{7} {(4 x - 1)}^{9} (4 \cdot 1 + \frac{d}{d x} [- 1]) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3.5

Multiply $4$ by $1$ .

$10 {(x + 5)}^{7} {(4 x - 1)}^{9} (4 + \frac{d}{d x} [- 1]) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3.6

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

$10 {(x + 5)}^{7} {(4 x - 1)}^{9} (4 + 0) + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $4$ and $0$ .

$10 {(x + 5)}^{7} {(4 x - 1)}^{9} \cdot 4 + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 3.7.2

Multiply $4$ by $10$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + {(4 x - 1)}^{10} \frac{d}{d x} [{(x + 5)}^{7}]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{7}$ and $g (x) = x + 5$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $x + 5$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + {(4 x - 1)}^{10} (\frac{d}{d u_{2}} [{u_{2}}^{7}] \frac{d}{d x} [x + 5])$

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 $n = 7$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + {(4 x - 1)}^{10} (7 {u_{2}}^{6} \frac{d}{d x} [x + 5])$

Step 4.3

Replace all occurrences of $u_{2}$ with $x + 5$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + {(4 x - 1)}^{10} (7 {(x + 5)}^{6} \frac{d}{d x} [x + 5])$

Step 5

Differentiate.

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Step 5.1

Move $7$ to the left of ${(4 x - 1)}^{10}$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + 7 \cdot {(4 x - 1)}^{10} {(x + 5)}^{6} \frac{d}{d x} [x + 5]$

Step 5.2

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + 7 {(4 x - 1)}^{10} {(x + 5)}^{6} (\frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 5.3

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

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + 7 {(4 x - 1)}^{10} {(x + 5)}^{6} (1 + \frac{d}{d x} [5])$

Step 5.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + 7 {(4 x - 1)}^{10} {(x + 5)}^{6} (1 + 0)$

Step 5.5

Simplify the expression.

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Step 5.5.1

Add $1$ and $0$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + 7 {(4 x - 1)}^{10} {(x + 5)}^{6} \cdot 1$

Step 5.5.2

Multiply $7$ by $1$ .

$40 {(x + 5)}^{7} {(4 x - 1)}^{9} + 7 {(4 x - 1)}^{10} {(x + 5)}^{6}$

Step 6

Simplify.

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Step 6.1

Factor ${(x + 5)}^{6} {(4 x - 1)}^{9}$ out of $40 {(x + 5)}^{7} {(4 x - 1)}^{9} + 7 {(4 x - 1)}^{10} {(x + 5)}^{6}$ .

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Step 6.1.1

Factor ${(x + 5)}^{6} {(4 x - 1)}^{9}$ out of $40 {(x + 5)}^{7} {(4 x - 1)}^{9}$ .

${(x + 5)}^{6} {(4 x - 1)}^{9} (40 (x + 5)) + 7 {(4 x - 1)}^{10} {(x + 5)}^{6}$

Step 6.1.2

Factor ${(x + 5)}^{6} {(4 x - 1)}^{9}$ out of $7 {(4 x - 1)}^{10} {(x + 5)}^{6}$ .

${(x + 5)}^{6} {(4 x - 1)}^{9} (40 (x + 5)) + {(x + 5)}^{6} {(4 x - 1)}^{9} (7 (4 x - 1))$

Step 6.1.3

Factor ${(x + 5)}^{6} {(4 x - 1)}^{9}$ out of ${(x + 5)}^{6} {(4 x - 1)}^{9} (40 (x + 5)) + {(x + 5)}^{6} {(4 x - 1)}^{9} (7 (4 x - 1))$ .

${(x + 5)}^{6} {(4 x - 1)}^{9} (40 (x + 5) + 7 (4 x - 1))$

Step 6.2

Reorder the factors of ${(x + 5)}^{6} {(4 x - 1)}^{9} (40 (x + 5) + 7 (4 x - 1))$ .

${(4 x - 1)}^{9} {(x + 5)}^{6} (40 (x + 5) + 7 (4 x - 1))$