Calculus Examples

$g (u) = \frac{3 u^{2}}{{(u^{2} + u)}^{3}}$

Step 1

Since $3$ is constant with respect to $u$ , the derivative of $\frac{3 u^{2}}{{(u^{2} + u)}^{3}}$ with respect to $u$ is $3 \frac{d}{d u} [\frac{u^{2}}{{(u^{2} + u)}^{3}}]$ .

$3 \frac{d}{d u} [\frac{u^{2}}{{(u^{2} + u)}^{3}}]$

Step 2

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(u^{2} + u)}^{3})}^{2}$ .

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Step 3.1.1

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

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

Step 3.1.2

Multiply $3$ by $2$ .

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

Step 3.2

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

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

Step 3.3

Move $2$ to the left of ${(u^{2} + u)}^{3}$ .

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

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d u} [f (g (u))]$ is $f' (g (u)) g' (u)$ where $f (u) = u^{3}$ and $g (u) = u^{2} + u$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate.

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Step 5.1

Multiply $3$ by $- 1$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

$3 \frac{2 {(u^{2} + u)}^{3} u - 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 5.5

Combine $3$ and $\frac{2 {(u^{2} + u)}^{3} u - 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1))}{{(u^{2} + u)}^{6}}$ .

$\frac{3 (2 {(u^{2} + u)}^{3} u - 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{3 (2 {(u^{2} + u)}^{3} u) + 3 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2

Simplify the numerator.

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Step 6.2.1

Factor $3 {(u^{2} + u)}^{2} u$ out of $3 \cdot 2 {(u^{2} + u)}^{3} u + 3 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))$ .

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Step 6.2.1.1

Factor $3 {(u^{2} + u)}^{2} u$ out of $3 \cdot 2 {(u^{2} + u)}^{3} u$ .

$\frac{3 {(u^{2} + u)}^{2} u (2 (u + 1) u) + 3 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))}{{(u^{2} + u)}^{6}}$

Step 6.2.1.2

Factor $3 {(u^{2} + u)}^{2} u$ out of $3 (- 3 u^{2} ({(u^{2} + u)}^{2} (2 u + 1)))$ .

$\frac{3 {(u^{2} + u)}^{2} u (2 (u + 1) u) + 3 {(u^{2} + u)}^{2} u (- 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.1.3

Factor $3 {(u^{2} + u)}^{2} u$ out of $3 {(u^{2} + u)}^{2} u (2 (u + 1) u) + 3 {(u^{2} + u)}^{2} u (- 3 u (2 u + 1))$ .

$\frac{3 {(u^{2} + u)}^{2} u (2 (u + 1) u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.2

Factor $u$ out of $u^{2} + u$ .

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Step 6.2.2.1

Factor $u$ out of $u^{2}$ .

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

Step 6.2.2.2

Raise $u$ to the power of $1$ .

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

Step 6.2.2.3

Factor $u$ out of $u^{1}$ .

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

Step 6.2.2.4

Factor $u$ out of $u \cdot u + u \cdot 1$ .

$\frac{3 {(u (u + 1))}^{2} u (2 (u + 1) u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.3

Apply the product rule to $u (u + 1)$ .

$\frac{3 u^{2} {(u + 1)}^{2} u (2 (u + 1) u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.4

Combine exponents.

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Step 6.2.4.1

Raise $u$ to the power of $1$ .

$\frac{3 (u^{1} u^{2}) {(u + 1)}^{2} (2 (u + 1) u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 u^{1 + 2} {(u + 1)}^{2} (2 (u + 1) u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.4.3

Add $1$ and $2$ .

$\frac{3 u^{3} {(u + 1)}^{2} (2 (u + 1) u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.5

Simplify each term.

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Step 6.2.5.1

Apply the distributive property.

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

Step 6.2.5.2

Multiply $2$ by $1$ .

$\frac{3 u^{3} {(u + 1)}^{2} ((2 u + 2) u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.5.3

Apply the distributive property.

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

Step 6.2.5.4

Multiply $u$ by $u$ by adding the exponents.

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Step 6.2.5.4.1

Move $u$ .

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

Step 6.2.5.4.2

Multiply $u$ by $u$ .

$\frac{3 u^{3} {(u + 1)}^{2} (2 u^{2} + 2 u - 3 u (2 u + 1))}{{(u^{2} + u)}^{6}}$

Step 6.2.5.5

Apply the distributive property.

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

Step 6.2.5.6

Rewrite using the commutative property of multiplication.

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

Step 6.2.5.7

Multiply $- 3$ by $1$ .

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

Step 6.2.5.8

Simplify each term.

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Step 6.2.5.8.1

Multiply $u$ by $u$ by adding the exponents.

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Step 6.2.5.8.1.1

Move $u$ .

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

Step 6.2.5.8.1.2

Multiply $u$ by $u$ .

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

Step 6.2.5.8.2

Multiply $- 3$ by $2$ .

$\frac{3 u^{3} {(u + 1)}^{2} (2 u^{2} + 2 u - 6 u^{2} - 3 u)}{{(u^{2} + u)}^{6}}$

Step 6.2.6

Subtract $6 u^{2}$ from $2 u^{2}$ .

$\frac{3 u^{3} {(u + 1)}^{2} (- 4 u^{2} + 2 u - 3 u)}{{(u^{2} + u)}^{6}}$

Step 6.2.7

Subtract $3 u$ from $2 u$ .

$\frac{3 u^{3} {(u + 1)}^{2} (- 4 u^{2} - u)}{{(u^{2} + u)}^{6}}$

Step 6.2.8

Factor $u$ out of $- 4 u^{2} - u$ .

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Step 6.2.8.1

Factor $u$ out of $- 4 u^{2}$ .

$\frac{3 u^{3} {(u + 1)}^{2} (u (- 4 u) - u)}{{(u^{2} + u)}^{6}}$

Step 6.2.8.2

Factor $u$ out of $- u$ .

$\frac{3 u^{3} {(u + 1)}^{2} (u (- 4 u) + u \cdot - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.8.3

Factor $u$ out of $u (- 4 u) + u \cdot - 1$ .

$\frac{3 u^{3} {(u + 1)}^{2} (u (- 4 u - 1))}{{(u^{2} + u)}^{6}}$

$\frac{3 u^{3} {(u + 1)}^{2} u (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.9

Combine exponents.

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Step 6.2.9.1

Raise $u$ to the power of $1$ .

$\frac{3 (u^{1} u^{3}) {(u + 1)}^{2} (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.9.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{3 u^{1 + 3} {(u + 1)}^{2} (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.2.9.3

Add $1$ and $3$ .

$\frac{3 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u^{2} + u)}^{6}}$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Factor $u$ out of $u^{2} + u$ .

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Step 6.3.1.1

Factor $u$ out of $u^{2}$ .

$\frac{3 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u \cdot u + u)}^{6}}$

Step 6.3.1.2

Raise $u$ to the power of $1$ .

$\frac{3 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u \cdot u + u^{1})}^{6}}$

Step 6.3.1.3

Factor $u$ out of $u^{1}$ .

$\frac{3 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u \cdot u + u \cdot 1)}^{6}}$

Step 6.3.1.4

Factor $u$ out of $u \cdot u + u \cdot 1$ .

$\frac{3 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{{(u (u + 1))}^{6}}$

Step 6.3.2

Apply the product rule to $u (u + 1)$ .

$\frac{3 u^{4} {(u + 1)}^{2} (- 4 u - 1)}{u^{6} {(u + 1)}^{6}}$

Step 6.4

Cancel the common factor of $u^{4}$ and $u^{6}$ .

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Step 6.4.1

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

$\frac{u^{4} (3 {(u + 1)}^{2} (- 4 u - 1))}{u^{6} {(u + 1)}^{6}}$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $u^{4}$ out of $u^{6} {(u + 1)}^{6}$ .

$\frac{u^{4} (3 {(u + 1)}^{2} (- 4 u - 1))}{u^{4} (u^{2} {(u + 1)}^{6})}$

Step 6.4.2.2

Cancel the common factor.

$\frac{u^{4} (3 {(u + 1)}^{2} (- 4 u - 1))}{u^{4} (u^{2} {(u + 1)}^{6})}$

Step 6.4.2.3

Rewrite the expression.

$\frac{3 {(u + 1)}^{2} (- 4 u - 1)}{u^{2} {(u + 1)}^{6}}$

Step 6.5

Cancel the common factor of ${(u + 1)}^{2}$ and ${(u + 1)}^{6}$ .

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Step 6.5.1

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

$\frac{{(u + 1)}^{2} (3 (- 4 u - 1))}{u^{2} {(u + 1)}^{6}}$

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Factor ${(u + 1)}^{2}$ out of $u^{2} {(u + 1)}^{6}$ .

$\frac{{(u + 1)}^{2} (3 (- 4 u - 1))}{{(u + 1)}^{2} (u^{2} {(u + 1)}^{4})}$

Step 6.5.2.2

Cancel the common factor.

$\frac{{(u + 1)}^{2} (3 (- 4 u - 1))}{{(u + 1)}^{2} (u^{2} {(u + 1)}^{4})}$

Step 6.5.2.3

Rewrite the expression.

$\frac{3 (- 4 u - 1)}{u^{2} {(u + 1)}^{4}}$

Step 6.6

Factor $- 1$ out of $- 4 u$ .

$\frac{3 (- (4 u) - 1)}{u^{2} {(u + 1)}^{4}}$

Step 6.7

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

$\frac{3 (- (4 u) - 1 (1))}{u^{2} {(u + 1)}^{4}}$

Step 6.8

Factor $- 1$ out of $- (4 u) - 1 (1)$ .

$\frac{3 (- (4 u + 1))}{u^{2} {(u + 1)}^{4}}$

Step 6.9

Rewrite $- (4 u + 1)$ as $- 1 (4 u + 1)$ .

$\frac{3 (- 1 (4 u + 1))}{u^{2} {(u + 1)}^{4}}$

Step 6.10

Move the negative in front of the fraction.

$- \frac{3 (4 u + 1)}{u^{2} {(u + 1)}^{4}}$