Calculus Examples

$y = {(4 x - 7)}^{5} {(5 x + 8)}^{3}$

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 $f (x) = {(4 x - 7)}^{5}$ and $g (x) = {(5 x + 8)}^{3}$ .

${(4 x - 7)}^{5} \frac{d}{d x} [{(5 x + 8)}^{3}] + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 5 x + 8$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $5 x + 8$ .

${(4 x - 7)}^{5} (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [5 x + 8]) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

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 $n = 3$ .

${(4 x - 7)}^{5} (3 {u_{1}}^{2} \frac{d}{d x} [5 x + 8]) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $5 x + 8$ .

${(4 x - 7)}^{5} (3 {(5 x + 8)}^{2} \frac{d}{d x} [5 x + 8]) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $5 x + 8$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [8]$ .

${(4 x - 7)}^{5} (3 {(5 x + 8)}^{2} (\frac{d}{d x} [5 x] + \frac{d}{d x} [8])) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 3.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

${(4 x - 7)}^{5} (3 {(5 x + 8)}^{2} (5 \frac{d}{d x} [x] + \frac{d}{d x} [8])) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 3.3

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

${(4 x - 7)}^{5} (3 {(5 x + 8)}^{2} (5 \cdot 1 + \frac{d}{d x} [8])) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 3.4

Multiply $5$ by $1$ .

${(4 x - 7)}^{5} (3 {(5 x + 8)}^{2} (5 + \frac{d}{d x} [8])) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 3.5

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

${(4 x - 7)}^{5} (3 {(5 x + 8)}^{2} (5 + 0)) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $5$ and $0$ .

${(4 x - 7)}^{5} (3 {(5 x + 8)}^{2} \cdot 5) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 3.6.2

Multiply $5$ by $3$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} \frac{d}{d x} [{(4 x - 7)}^{5}]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = 4 x - 7$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $4 x - 7$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [4 x - 7])$

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 $n = 5$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {u_{2}}^{4} \frac{d}{d x} [4 x - 7])$

Step 4.3

Replace all occurrences of $u_{2}$ with $4 x - 7$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {(4 x - 7)}^{4} \frac{d}{d x} [4 x - 7])$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $4 x - 7$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7]$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {(4 x - 7)}^{4} (\frac{d}{d x} [4 x] + \frac{d}{d x} [- 7]))$

Step 5.2

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

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {(4 x - 7)}^{4} (4 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]))$

Step 5.3

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

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {(4 x - 7)}^{4} (4 \cdot 1 + \frac{d}{d x} [- 7]))$

Step 5.4

Multiply $4$ by $1$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {(4 x - 7)}^{4} (4 + \frac{d}{d x} [- 7]))$

Step 5.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {(4 x - 7)}^{4} (4 + 0))$

Step 5.6

Simplify the expression.

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Step 5.6.1

Add $4$ and $0$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (5 {(4 x - 7)}^{4} \cdot 4)$

Step 5.6.2

Multiply $4$ by $5$ .

${(4 x - 7)}^{5} (15 {(5 x + 8)}^{2}) + {(5 x + 8)}^{3} (20 {(4 x - 7)}^{4})$

Step 6

Simplify.

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Step 6.1

Move $15$ to the left of ${(4 x - 7)}^{5}$ .

$15 \cdot {(4 x - 7)}^{5} {(5 x + 8)}^{2} + {(5 x + 8)}^{3} (20 {(4 x - 7)}^{4})$

Step 6.2

Move $20$ to the left of ${(5 x + 8)}^{3}$ .

$15 {(4 x - 7)}^{5} {(5 x + 8)}^{2} + 20 \cdot {(5 x + 8)}^{3} {(4 x - 7)}^{4}$

Step 6.3

Factor $5 {(4 x - 7)}^{4} {(5 x + 8)}^{2}$ out of $15 {(4 x - 7)}^{5} {(5 x + 8)}^{2} + 20 {(5 x + 8)}^{3} {(4 x - 7)}^{4}$ .

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Step 6.3.1

Factor $5 {(4 x - 7)}^{4} {(5 x + 8)}^{2}$ out of $15 {(4 x - 7)}^{5} {(5 x + 8)}^{2}$ .

$5 {(4 x - 7)}^{4} {(5 x + 8)}^{2} (3 (4 x - 7)) + 20 {(5 x + 8)}^{3} {(4 x - 7)}^{4}$

Step 6.3.2

Factor $5 {(4 x - 7)}^{4} {(5 x + 8)}^{2}$ out of $20 {(5 x + 8)}^{3} {(4 x - 7)}^{4}$ .

$5 {(4 x - 7)}^{4} {(5 x + 8)}^{2} (3 (4 x - 7)) + 5 {(4 x - 7)}^{4} {(5 x + 8)}^{2} (4 (5 x + 8))$

Step 6.3.3

Factor $5 {(4 x - 7)}^{4} {(5 x + 8)}^{2}$ out of $5 {(4 x - 7)}^{4} {(5 x + 8)}^{2} (3 (4 x - 7)) + 5 {(4 x - 7)}^{4} {(5 x + 8)}^{2} (4 (5 x + 8))$ .

$5 {(4 x - 7)}^{4} {(5 x + 8)}^{2} (3 (4 x - 7) + 4 (5 x + 8))$