Calculus Examples

$y = {(8 + x^{5})}^{4} - {(6 + x^{7})}^{5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} ({(8 + x^{5})}^{4} - {(6 + x^{7})}^{5})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of ${(8 + x^{5})}^{4} - {(6 + x^{7})}^{5}$ with respect to $x$ is $\frac{d}{d x} [{(8 + x^{5})}^{4}] + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$ .

$\frac{d}{d x} [{(8 + x^{5})}^{4}] + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2

Evaluate $\frac{d}{d x} [{(8 + x^{5})}^{4}]$ .

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Step 3.2.1

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 $g (x) = 8 + x^{5}$ .

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Step 3.2.1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{4}] \frac{d}{d x} [8 + x^{5}] + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2.1.2

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

$4 {u_{1}}^{3} \frac{d}{d x} [8 + x^{5}] + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2.1.3

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

$4 {(8 + x^{5})}^{3} \frac{d}{d x} [8 + x^{5}] + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2.2

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

$4 {(8 + x^{5})}^{3} (\frac{d}{d x} [8] + \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2.3

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

$4 {(8 + x^{5})}^{3} (0 + \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$4 {(8 + x^{5})}^{3} (0 + 5 x^{4}) + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2.5

Add $0$ and $5 x^{4}$ .

$4 {(8 + x^{5})}^{3} (5 x^{4}) + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.2.6

Multiply $5$ by $4$ .

$20 {(8 + x^{5})}^{3} x^{4} + \frac{d}{d x} [- {(6 + x^{7})}^{5}]$

Step 3.3

Evaluate $\frac{d}{d x} [- {(6 + x^{7})}^{5}]$ .

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Step 3.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(6 + x^{7})}^{5}$ with respect to $x$ is $- \frac{d}{d x} [{(6 + x^{7})}^{5}]$ .

$20 {(8 + x^{5})}^{3} x^{4} - \frac{d}{d x} [{(6 + x^{7})}^{5}]$

Step 3.3.2

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 $g (x) = 6 + x^{7}$ .

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Step 3.3.2.1

To apply the Chain Rule, set $u_{2}$ as $6 + x^{7}$ .

$20 {(8 + x^{5})}^{3} x^{4} - (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [6 + x^{7}])$

Step 3.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$20 {(8 + x^{5})}^{3} x^{4} - (5 {u_{2}}^{4} \frac{d}{d x} [6 + x^{7}])$

Step 3.3.2.3

Replace all occurrences of $u_{2}$ with $6 + x^{7}$ .

$20 {(8 + x^{5})}^{3} x^{4} - (5 {(6 + x^{7})}^{4} \frac{d}{d x} [6 + x^{7}])$

Step 3.3.3

By the Sum Rule, the derivative of $6 + x^{7}$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [x^{7}]$ .

$20 {(8 + x^{5})}^{3} x^{4} - (5 {(6 + x^{7})}^{4} (\frac{d}{d x} [6] + \frac{d}{d x} [x^{7}]))$

Step 3.3.4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$20 {(8 + x^{5})}^{3} x^{4} - (5 {(6 + x^{7})}^{4} (0 + \frac{d}{d x} [x^{7}]))$

Step 3.3.5

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

$20 {(8 + x^{5})}^{3} x^{4} - (5 {(6 + x^{7})}^{4} (0 + 7 x^{6}))$

Step 3.3.6

Add $0$ and $7 x^{6}$ .

$20 {(8 + x^{5})}^{3} x^{4} - (5 {(6 + x^{7})}^{4} (7 x^{6}))$

Step 3.3.7

Multiply $7$ by $5$ .

$20 {(8 + x^{5})}^{3} x^{4} - (35 {(6 + x^{7})}^{4} x^{6})$

Step 3.3.8

Multiply $35$ by $- 1$ .

$20 {(8 + x^{5})}^{3} x^{4} - 35 {(6 + x^{7})}^{4} x^{6}$

Step 3.4

Factor $5 x^{4}$ out of $20 {(8 + x^{5})}^{3} x^{4} - 35 {(6 + x^{7})}^{4} x^{6}$ .

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Step 3.4.1

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

$5 x^{4} (4 {(8 + x^{5})}^{3}) - 35 {(6 + x^{7})}^{4} x^{6}$

Step 3.4.2

Factor $5 x^{4}$ out of $- 35 {(6 + x^{7})}^{4} x^{6}$ .

$5 x^{4} (4 {(8 + x^{5})}^{3}) + 5 x^{4} (- 7 {(6 + x^{7})}^{4} x^{2})$

Step 3.4.3

Factor $5 x^{4}$ out of $5 x^{4} (4 {(8 + x^{5})}^{3}) + 5 x^{4} (- 7 {(6 + x^{7})}^{4} x^{2})$ .

$5 x^{4} (4 {(8 + x^{5})}^{3} - 7 {(6 + x^{7})}^{4} x^{2})$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 5 x^{4} (4 {(8 + x^{5})}^{3} - 7 {(6 + x^{7})}^{4} x^{2})$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 5 x^{4} (4 {(8 + x^{5})}^{3} - 7 {(6 + x^{7})}^{4} x^{2})$