Calculus Examples

$f (x) = \frac{{(3 x - 5)}^{3}}{{(2 x^{2} + 1)}^{4}}$

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

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

Step 2

Multiply the exponents in ${({(2 x^{2} + 1)}^{4})}^{2}$ .

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Step 2.1

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

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

Step 2.2

Multiply $4$ by $2$ .

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

Step 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 $g (x) = 3 x - 5$ .

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Step 3.1

To apply the Chain Rule, set $u_{1}$ as $3 x - 5$ .

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

Step 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 $n = 3$ .

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

Step 3.3

Replace all occurrences of $u_{1}$ with $3 x - 5$ .

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

Step 4

Differentiate.

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Step 4.1

Move $3$ to the left of ${(2 x^{2} + 1)}^{4}$ .

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

Step 4.2

By the Sum Rule, the derivative of $3 x - 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Multiply $3$ by $1$ .

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

Step 4.6

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

$\frac{3 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} (3 + 0) - {(3 x - 5)}^{3} \frac{d}{d x} [{(2 x^{2} + 1)}^{4}]}{{(2 x^{2} + 1)}^{8}}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Add $3$ and $0$ .

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

Step 4.7.2

Multiply $3$ by $3$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - {(3 x - 5)}^{3} \frac{d}{d x} [{(2 x^{2} + 1)}^{4}]}{{(2 x^{2} + 1)}^{8}}$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{4}$ and $g (x) = 2 x^{2} + 1$ .

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Step 5.1

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

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - {(3 x - 5)}^{3} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [2 x^{2} + 1])}{{(2 x^{2} + 1)}^{8}}$

Step 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 $n = 4$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - {(3 x - 5)}^{3} (4 {u_{2}}^{3} \frac{d}{d x} [2 x^{2} + 1])}{{(2 x^{2} + 1)}^{8}}$

Step 5.3

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

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - {(3 x - 5)}^{3} (4 {(2 x^{2} + 1)}^{3} \frac{d}{d x} [2 x^{2} + 1])}{{(2 x^{2} + 1)}^{8}}$

Step 6

Differentiate.

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Step 6.1

Multiply $4$ by $- 1$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} \frac{d}{d x} [2 x^{2} + 1])}{{(2 x^{2} + 1)}^{8}}$

Step 6.2

By the Sum Rule, the derivative of $2 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} (\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [1]))}{{(2 x^{2} + 1)}^{8}}$

Step 6.3

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

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} (2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]))}{{(2 x^{2} + 1)}^{8}}$

Step 6.4

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

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} (2 (2 x) + \frac{d}{d x} [1]))}{{(2 x^{2} + 1)}^{8}}$

Step 6.5

Multiply $2$ by $2$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} (4 x + \frac{d}{d x} [1]))}{{(2 x^{2} + 1)}^{8}}$

Step 6.6

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

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} (4 x + 0))}{{(2 x^{2} + 1)}^{8}}$

Step 6.7

Simplify the expression.

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Step 6.7.1

Add $4 x$ and $0$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} (4 x))}{{(2 x^{2} + 1)}^{8}}$

Step 6.7.2

Move $4$ to the left of ${(2 x^{2} + 1)}^{3}$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 4 {(3 x - 5)}^{3} (4 \cdot {(2 x^{2} + 1)}^{3} x)}{{(2 x^{2} + 1)}^{8}}$

Step 6.7.3

Multiply $4$ by $- 4$ .

$\frac{9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 16 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} x)}{{(2 x^{2} + 1)}^{8}}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Factor ${(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2}$ out of $9 {(2 x^{2} + 1)}^{4} {(3 x - 5)}^{2} - 16 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} x)$ .

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Step 7.1.1.1

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

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (9 (2 x^{2} + 1)) - 16 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.1.2

Factor ${(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2}$ out of $- 16 {(3 x - 5)}^{3} ({(2 x^{2} + 1)}^{3} x)$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (9 (2 x^{2} + 1)) + {(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (- 16 (3 x - 5) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.1.3

Factor ${(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2}$ out of ${(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (9 (2 x^{2} + 1)) + {(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (- 16 (3 x - 5) x)$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (9 (2 x^{2} + 1) - 16 (3 x - 5) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.2

Apply the distributive property.

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (9 (2 x^{2}) + 9 \cdot 1 - 16 (3 x - 5) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.3

Multiply $2$ by $9$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 \cdot 1 - 16 (3 x - 5) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.4

Multiply $9$ by $1$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 - 16 (3 x - 5) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.5

Apply the distributive property.

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 + (- 16 (3 x) - 16 \cdot - 5) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.6

Multiply $3$ by $- 16$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 + (- 48 x - 16 \cdot - 5) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.7

Multiply $- 16$ by $- 5$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 + (- 48 x + 80) x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.8

Apply the distributive property.

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 - 48 x \cdot x + 80 x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.9

Multiply $x$ by $x$ by adding the exponents.

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Step 7.1.9.1

Move $x$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 - 48 (x \cdot x) + 80 x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.9.2

Multiply $x$ by $x$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (18 x^{2} + 9 - 48 x^{2} + 80 x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.10

Subtract $48 x^{2}$ from $18 x^{2}$ .

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (- 30 x^{2} + 9 + 80 x)}{{(2 x^{2} + 1)}^{8}}$

Step 7.1.11

Reorder terms.

$\frac{{(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (- 30 x^{2} + 80 x + 9)}{{(2 x^{2} + 1)}^{8}}$

Step 7.2

Cancel the common factor of ${(2 x^{2} + 1)}^{3}$ and ${(2 x^{2} + 1)}^{8}$ .

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Step 7.2.1

Factor ${(2 x^{2} + 1)}^{3}$ out of ${(2 x^{2} + 1)}^{3} {(3 x - 5)}^{2} (- 30 x^{2} + 80 x + 9)$ .

$\frac{{(2 x^{2} + 1)}^{3} ({(3 x - 5)}^{2} (- 30 x^{2} + 80 x + 9))}{{(2 x^{2} + 1)}^{8}}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor ${(2 x^{2} + 1)}^{3}$ out of ${(2 x^{2} + 1)}^{8}$ .

$\frac{{(2 x^{2} + 1)}^{3} ({(3 x - 5)}^{2} (- 30 x^{2} + 80 x + 9))}{{(2 x^{2} + 1)}^{3} {(2 x^{2} + 1)}^{5}}$

Step 7.2.2.2

Cancel the common factor.

$\frac{{(2 x^{2} + 1)}^{3} ({(3 x - 5)}^{2} (- 30 x^{2} + 80 x + 9))}{{(2 x^{2} + 1)}^{3} {(2 x^{2} + 1)}^{5}}$

Step 7.2.2.3

Rewrite the expression.

$\frac{{(3 x - 5)}^{2} (- 30 x^{2} + 80 x + 9)}{{(2 x^{2} + 1)}^{5}}$

Step 7.3

Factor $- 1$ out of $- 30 x^{2}$ .

$\frac{{(3 x - 5)}^{2} (- (30 x^{2}) + 80 x + 9)}{{(2 x^{2} + 1)}^{5}}$

Step 7.4

Factor $- 1$ out of $80 x$ .

$\frac{{(3 x - 5)}^{2} (- (30 x^{2}) - (- 80 x) + 9)}{{(2 x^{2} + 1)}^{5}}$

Step 7.5

Factor $- 1$ out of $- (30 x^{2}) - (- 80 x)$ .

$\frac{{(3 x - 5)}^{2} (- (30 x^{2} - 80 x) + 9)}{{(2 x^{2} + 1)}^{5}}$

Step 7.6

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{{(3 x - 5)}^{2} (- (30 x^{2} - 80 x) - 1 (- 9))}{{(2 x^{2} + 1)}^{5}}$

Step 7.7

Factor $- 1$ out of $- (30 x^{2} - 80 x) - 1 (- 9)$ .

$\frac{{(3 x - 5)}^{2} (- (30 x^{2} - 80 x - 9))}{{(2 x^{2} + 1)}^{5}}$

Step 7.8

Rewrite $- (30 x^{2} - 80 x - 9)$ as $- 1 (30 x^{2} - 80 x - 9)$ .

$\frac{{(3 x - 5)}^{2} (- 1 (30 x^{2} - 80 x - 9))}{{(2 x^{2} + 1)}^{5}}$

Step 7.9

Move the negative in front of the fraction.

$- \frac{{(3 x - 5)}^{2} (30 x^{2} - 80 x - 9)}{{(2 x^{2} + 1)}^{5}}$