Calculus Examples

$y = \frac{{(x + 5)}^{3}}{{(x - 3)}^{3}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{{(x + 5)}^{3}}{{(x - 3)}^{3}})$

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

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = {(x + 5)}^{3}$ and $g (x) = {(x - 3)}^{3}$ .

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

Step 3.2

Multiply the exponents in ${({(x - 3)}^{3})}^{2}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.2

Multiply $3$ by $2$ .

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

Step 3.3

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 $g (x) = x + 5$ .

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Step 3.3.1

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

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

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

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

Step 3.3.3

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

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

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

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

Step 3.4.3

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

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

Step 3.4.4

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

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

Step 3.4.5

Simplify the expression.

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Step 3.4.5.1

Add $1$ and $0$ .

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

Step 3.4.5.2

Multiply $3$ by $1$ .

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

Step 3.5

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 $g (x) = x - 3$ .

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Step 3.5.1

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

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

Step 3.5.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 3$ .

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

Step 3.5.3

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

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

Step 3.6

Differentiate.

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Step 3.6.1

Multiply $3$ by $- 1$ .

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3.6.5

Simplify the expression.

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Step 3.6.5.1

Add $1$ and $0$ .

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

Step 3.6.5.2

Multiply ${(x - 3)}^{2}$ by $1$ .

$\frac{3 {(x - 3)}^{3} {(x + 5)}^{2} - 3 {(x + 5)}^{3} {(x - 3)}^{2}}{{(x - 3)}^{6}}$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

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

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Step 3.7.1.1.1

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

$\frac{3 {(x - 3)}^{2} {(x + 5)}^{2} (x - 3) - 3 {(x + 5)}^{3} {(x - 3)}^{2}}{{(x - 3)}^{6}}$

Step 3.7.1.1.2

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

$\frac{3 {(x - 3)}^{2} {(x + 5)}^{2} (x - 3) + 3 {(x - 3)}^{2} {(x + 5)}^{2} (- (x + 5))}{{(x - 3)}^{6}}$

Step 3.7.1.1.3

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

$\frac{3 {(x - 3)}^{2} {(x + 5)}^{2} (x - 3 - (x + 5))}{{(x - 3)}^{6}}$

Step 3.7.1.2

Apply the distributive property.

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

Step 3.7.1.3

Multiply $- 1$ by $5$ .

$\frac{3 {(x - 3)}^{2} {(x + 5)}^{2} (x - 3 - x - 5)}{{(x - 3)}^{6}}$

Step 3.7.1.4

Subtract $x$ from $x$ .

$\frac{3 {(x - 3)}^{2} {(x + 5)}^{2} (0 - 3 - 5)}{{(x - 3)}^{6}}$

Step 3.7.1.5

Subtract $3$ from $0$ .

$\frac{3 {(x - 3)}^{2} {(x + 5)}^{2} (- 3 - 5)}{{(x - 3)}^{6}}$

Step 3.7.1.6

Subtract $5$ from $- 3$ .

$\frac{3 {(x - 3)}^{2} {(x + 5)}^{2} \cdot - 8}{{(x - 3)}^{6}}$

Step 3.7.1.7

Multiply $- 8$ by $3$ .

$\frac{- 24 {(x - 3)}^{2} {(x + 5)}^{2}}{{(x - 3)}^{6}}$

Step 3.7.2

Combine terms.

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Step 3.7.2.1

Cancel the common factor of ${(x - 3)}^{2}$ and ${(x - 3)}^{6}$ .

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Step 3.7.2.1.1

Factor ${(x - 3)}^{2}$ out of $- 24 {(x - 3)}^{2} {(x + 5)}^{2}$ .

$\frac{{(x - 3)}^{2} (- 24 {(x + 5)}^{2})}{{(x - 3)}^{6}}$

Step 3.7.2.1.2

Cancel the common factors.

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Step 3.7.2.1.2.1

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

$\frac{{(x - 3)}^{2} (- 24 {(x + 5)}^{2})}{{(x - 3)}^{2} {(x - 3)}^{4}}$

Step 3.7.2.1.2.2

Cancel the common factor.

$\frac{{(x - 3)}^{2} (- 24 {(x + 5)}^{2})}{{(x - 3)}^{2} {(x - 3)}^{4}}$

Step 3.7.2.1.2.3

Rewrite the expression.

$\frac{- 24 {(x + 5)}^{2}}{{(x - 3)}^{4}}$

Step 3.7.2.2

Move the negative in front of the fraction.

$- \frac{24 {(x + 5)}^{2}}{{(x - 3)}^{4}}$

Step 4

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

$y' = - \frac{24 {(x + 5)}^{2}}{{(x - 3)}^{4}}$

Step 5

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

$\frac{d y}{d x} = - \frac{24 {(x + 5)}^{2}}{{(x - 3)}^{4}}$