Calculus Examples

$y = \frac{300 x + 300}{{(2 x + 3)}^{2}}$

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 $f (x) = 300 x + 300$ and $g (x) = {(2 x + 3)}^{2}$ .

$\frac{{(2 x + 3)}^{2} \frac{d}{d x} [300 x + 300] - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{({(2 x + 3)}^{2})}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

$\frac{{(2 x + 3)}^{2} \frac{d}{d x} [300 x + 300] - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{2 \cdot 2}}$

Step 2.1.2

Multiply $2$ by $2$ .

$\frac{{(2 x + 3)}^{2} \frac{d}{d x} [300 x + 300] - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 2.2

By the Sum Rule, the derivative of $300 x + 300$ with respect to $x$ is $\frac{d}{d x} [300 x] + \frac{d}{d x} [300]$ .

$\frac{{(2 x + 3)}^{2} (\frac{d}{d x} [300 x] + \frac{d}{d x} [300]) - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 2.3

Since $300$ is constant with respect to $x$ , the derivative of $300 x$ with respect to $x$ is $300 \frac{d}{d x} [x]$ .

$\frac{{(2 x + 3)}^{2} (300 \frac{d}{d x} [x] + \frac{d}{d x} [300]) - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 2.4

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

$\frac{{(2 x + 3)}^{2} (300 \cdot 1 + \frac{d}{d x} [300]) - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 2.5

Multiply $300$ by $1$ .

$\frac{{(2 x + 3)}^{2} (300 + \frac{d}{d x} [300]) - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 2.6

Since $300$ is constant with respect to $x$ , the derivative of $300$ with respect to $x$ is $0$ .

$\frac{{(2 x + 3)}^{2} (300 + 0) - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Add $300$ and $0$ .

$\frac{{(2 x + 3)}^{2} \cdot 300 - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 2.7.2

Move $300$ to the left of ${(2 x + 3)}^{2}$ .

$\frac{300 \cdot {(2 x + 3)}^{2} - (300 x + 300) \frac{d}{d x} [{(2 x + 3)}^{2}]}{{(2 x + 3)}^{4}}$

Step 3

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

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Step 3.1

To apply the Chain Rule, set $u$ as $2 x + 3$ .

$\frac{300 {(2 x + 3)}^{2} - (300 x + 300) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [2 x + 3])}{{(2 x + 3)}^{4}}$

Step 3.2

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

$\frac{300 {(2 x + 3)}^{2} - (300 x + 300) (2 u \frac{d}{d x} [2 x + 3])}{{(2 x + 3)}^{4}}$

Step 3.3

Replace all occurrences of $u$ with $2 x + 3$ .

$\frac{300 {(2 x + 3)}^{2} - (300 x + 300) (2 (2 x + 3) \frac{d}{d x} [2 x + 3])}{{(2 x + 3)}^{4}}$

Step 4

Differentiate.

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Step 4.1

Multiply $2$ by $- 1$ .

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) ((2 x + 3) \frac{d}{d x} [2 x + 3])}{{(2 x + 3)}^{4}}$

Step 4.2

By the Sum Rule, the derivative of $2 x + 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$ .

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) ((2 x + 3) (\frac{d}{d x} [2 x] + \frac{d}{d x} [3]))}{{(2 x + 3)}^{4}}$

Step 4.3

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

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) ((2 x + 3) (2 \frac{d}{d x} [x] + \frac{d}{d x} [3]))}{{(2 x + 3)}^{4}}$

Step 4.4

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

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) ((2 x + 3) (2 \cdot 1 + \frac{d}{d x} [3]))}{{(2 x + 3)}^{4}}$

Step 4.5

Multiply $2$ by $1$ .

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) ((2 x + 3) (2 + \frac{d}{d x} [3]))}{{(2 x + 3)}^{4}}$

Step 4.6

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

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) ((2 x + 3) (2 + 0))}{{(2 x + 3)}^{4}}$

Step 4.7

Simplify the expression.

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Step 4.7.1

Add $2$ and $0$ .

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) ((2 x + 3) \cdot 2)}{{(2 x + 3)}^{4}}$

Step 4.7.2

Move $2$ to the left of $2 x + 3$ .

$\frac{300 {(2 x + 3)}^{2} - 2 (300 x + 300) (2 \cdot (2 x + 3))}{{(2 x + 3)}^{4}}$

Step 4.7.3

Multiply $2$ by $- 2$ .

$\frac{300 {(2 x + 3)}^{2} - 4 (300 x + 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{300 {(2 x + 3)}^{2} + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Rewrite ${(2 x + 3)}^{2}$ as $(2 x + 3) (2 x + 3)$ .

$\frac{300 ((2 x + 3) (2 x + 3)) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.2

Expand $(2 x + 3) (2 x + 3)$ using the FOIL Method.

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Step 5.2.1.2.1

Apply the distributive property.

$\frac{300 (2 x (2 x + 3) + 3 (2 x + 3)) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.2.2

Apply the distributive property.

$\frac{300 (2 x (2 x) + 2 x \cdot 3 + 3 (2 x + 3)) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.2.3

Apply the distributive property.

$\frac{300 (2 x (2 x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3

Simplify and combine like terms.

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Step 5.2.1.3.1

Simplify each term.

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Step 5.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{300 (2 \cdot 2 x \cdot x + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3.1.2

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

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Step 5.2.1.3.1.2.1

Move $x$ .

$\frac{300 (2 \cdot 2 (x \cdot x) + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3.1.2.2

Multiply $x$ by $x$ .

$\frac{300 (2 \cdot 2 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3.1.3

Multiply $2$ by $2$ .

$\frac{300 (4 x^{2} + 2 x \cdot 3 + 3 (2 x) + 3 \cdot 3) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3.1.4

Multiply $3$ by $2$ .

$\frac{300 (4 x^{2} + 6 x + 3 (2 x) + 3 \cdot 3) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3.1.5

Multiply $2$ by $3$ .

$\frac{300 (4 x^{2} + 6 x + 6 x + 3 \cdot 3) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3.1.6

Multiply $3$ by $3$ .

$\frac{300 (4 x^{2} + 6 x + 6 x + 9) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.3.2

Add $6 x$ and $6 x$ .

$\frac{300 (4 x^{2} + 12 x + 9) + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.4

Apply the distributive property.

$\frac{300 (4 x^{2}) + 300 (12 x) + 300 \cdot 9 + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.5

Simplify.

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Step 5.2.1.5.1

Multiply $4$ by $300$ .

$\frac{1200 x^{2} + 300 (12 x) + 300 \cdot 9 + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.5.2

Multiply $12$ by $300$ .

$\frac{1200 x^{2} + 3600 x + 300 \cdot 9 + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.5.3

Multiply $300$ by $9$ .

$\frac{1200 x^{2} + 3600 x + 2700 + (- 4 (300 x) - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.6

Simplify each term.

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Step 5.2.1.6.1

Multiply $300$ by $- 4$ .

$\frac{1200 x^{2} + 3600 x + 2700 + (- 1200 x - 4 \cdot 300) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.6.2

Multiply $- 4$ by $300$ .

$\frac{1200 x^{2} + 3600 x + 2700 + (- 1200 x - 1200) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.7

Expand $(- 1200 x - 1200) (2 x + 3)$ using the FOIL Method.

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Step 5.2.1.7.1

Apply the distributive property.

$\frac{1200 x^{2} + 3600 x + 2700 - 1200 x (2 x + 3) - 1200 (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.7.2

Apply the distributive property.

$\frac{1200 x^{2} + 3600 x + 2700 - 1200 x (2 x) - 1200 x \cdot 3 - 1200 (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.2.1.7.3

Apply the distributive property.

$\frac{1200 x^{2} + 3600 x + 2700 - 1200 x (2 x) - 1200 x \cdot 3 - 1200 (2 x) - 1200 \cdot 3}{{(2 x + 3)}^{4}}$

Step 5.2.1.8

Simplify and combine like terms.

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Step 5.2.1.8.1

Simplify each term.

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Step 5.2.1.8.1.1

Rewrite using the commutative property of multiplication.

$\frac{1200 x^{2} + 3600 x + 2700 - 1200 \cdot 2 x \cdot x - 1200 x \cdot 3 - 1200 (2 x) - 1200 \cdot 3}{{(2 x + 3)}^{4}}$

Step 5.2.1.8.1.2

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

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Step 5.2.1.8.1.2.1

Move $x$ .

$\frac{1200 x^{2} + 3600 x + 2700 - 1200 \cdot 2 (x \cdot x) - 1200 x \cdot 3 - 1200 (2 x) - 1200 \cdot 3}{{(2 x + 3)}^{4}}$

Step 5.2.1.8.1.2.2

Multiply $x$ by $x$ .

$\frac{1200 x^{2} + 3600 x + 2700 - 1200 \cdot 2 x^{2} - 1200 x \cdot 3 - 1200 (2 x) - 1200 \cdot 3}{{(2 x + 3)}^{4}}$

Step 5.2.1.8.1.3

Multiply $- 1200$ by $2$ .

$\frac{1200 x^{2} + 3600 x + 2700 - 2400 x^{2} - 1200 x \cdot 3 - 1200 (2 x) - 1200 \cdot 3}{{(2 x + 3)}^{4}}$

Step 5.2.1.8.1.4

Multiply $3$ by $- 1200$ .

$\frac{1200 x^{2} + 3600 x + 2700 - 2400 x^{2} - 3600 x - 1200 (2 x) - 1200 \cdot 3}{{(2 x + 3)}^{4}}$

Step 5.2.1.8.1.5

Multiply $2$ by $- 1200$ .

$\frac{1200 x^{2} + 3600 x + 2700 - 2400 x^{2} - 3600 x - 2400 x - 1200 \cdot 3}{{(2 x + 3)}^{4}}$

Step 5.2.1.8.1.6

Multiply $- 1200$ by $3$ .

$\frac{1200 x^{2} + 3600 x + 2700 - 2400 x^{2} - 3600 x - 2400 x - 3600}{{(2 x + 3)}^{4}}$

Step 5.2.1.8.2

Subtract $2400 x$ from $- 3600 x$ .

$\frac{1200 x^{2} + 3600 x + 2700 - 2400 x^{2} - 6000 x - 3600}{{(2 x + 3)}^{4}}$

Step 5.2.2

Subtract $2400 x^{2}$ from $1200 x^{2}$ .

$\frac{- 1200 x^{2} + 3600 x + 2700 - 6000 x - 3600}{{(2 x + 3)}^{4}}$

Step 5.2.3

Subtract $6000 x$ from $3600 x$ .

$\frac{- 1200 x^{2} - 2400 x + 2700 - 3600}{{(2 x + 3)}^{4}}$

Step 5.2.4

Subtract $3600$ from $2700$ .

$\frac{- 1200 x^{2} - 2400 x - 900}{{(2 x + 3)}^{4}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Factor $300$ out of $- 1200 x^{2} - 2400 x - 900$ .

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Step 5.3.1.1

Factor $300$ out of $- 1200 x^{2}$ .

$\frac{300 (- 4 x^{2}) - 2400 x - 900}{{(2 x + 3)}^{4}}$

Step 5.3.1.2

Factor $300$ out of $- 2400 x$ .

$\frac{300 (- 4 x^{2}) + 300 (- 8 x) - 900}{{(2 x + 3)}^{4}}$

Step 5.3.1.3

Factor $300$ out of $- 900$ .

$\frac{300 (- 4 x^{2}) + 300 (- 8 x) + 300 (- 3)}{{(2 x + 3)}^{4}}$

Step 5.3.1.4

Factor $300$ out of $300 (- 4 x^{2}) + 300 (- 8 x)$ .

$\frac{300 (- 4 x^{2} - 8 x) + 300 (- 3)}{{(2 x + 3)}^{4}}$

Step 5.3.1.5

Factor $300$ out of $300 (- 4 x^{2} - 8 x) + 300 (- 3)$ .

$\frac{300 (- 4 x^{2} - 8 x - 3)}{{(2 x + 3)}^{4}}$

Step 5.3.2

Factor by grouping.

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Step 5.3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 4 \cdot - 3 = 12$ and whose sum is $b = - 8$ .

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Step 5.3.2.1.1

Factor $- 8$ out of $- 8 x$ .

$\frac{300 (- 4 x^{2} - 8 (x) - 3)}{{(2 x + 3)}^{4}}$

Step 5.3.2.1.2

Rewrite $- 8$ as $- 2$ plus $- 6$

$\frac{300 (- 4 x^{2} + (- 2 - 6) x - 3)}{{(2 x + 3)}^{4}}$

Step 5.3.2.1.3

Apply the distributive property.

$\frac{300 (- 4 x^{2} - 2 x - 6 x - 3)}{{(2 x + 3)}^{4}}$

Step 5.3.2.2

Factor out the greatest common factor from each group.

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Step 5.3.2.2.1

Group the first two terms and the last two terms.

$\frac{300 ((- 4 x^{2} - 2 x) - 6 x - 3)}{{(2 x + 3)}^{4}}$

Step 5.3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{300 (2 x (- 2 x - 1) + 3 (- 2 x - 1))}{{(2 x + 3)}^{4}}$

Step 5.3.2.3

Factor the polynomial by factoring out the greatest common factor, $- 2 x - 1$ .

$\frac{300 ((- 2 x - 1) (2 x + 3))}{{(2 x + 3)}^{4}}$

$\frac{300 (- 2 x - 1) (2 x + 3)}{{(2 x + 3)}^{4}}$

Step 5.4

Cancel the common factor of $2 x + 3$ and ${(2 x + 3)}^{4}$ .

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Step 5.4.1

Factor $2 x + 3$ out of $300 (- 2 x - 1) (2 x + 3)$ .

$\frac{(2 x + 3) (300 (- 2 x - 1))}{{(2 x + 3)}^{4}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $2 x + 3$ out of ${(2 x + 3)}^{4}$ .

$\frac{(2 x + 3) (300 (- 2 x - 1))}{(2 x + 3) {(2 x + 3)}^{3}}$

Step 5.4.2.2

Cancel the common factor.

$\frac{(2 x + 3) (300 (- 2 x - 1))}{(2 x + 3) {(2 x + 3)}^{3}}$

Step 5.4.2.3

Rewrite the expression.

$\frac{300 (- 2 x - 1)}{{(2 x + 3)}^{3}}$

Step 5.5

Factor $- 1$ out of $- 2 x$ .

$\frac{300 (- (2 x) - 1)}{{(2 x + 3)}^{3}}$

Step 5.6

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

$\frac{300 (- (2 x) - 1 (1))}{{(2 x + 3)}^{3}}$

Step 5.7

Factor $- 1$ out of $- (2 x) - 1 (1)$ .

$\frac{300 (- (2 x + 1))}{{(2 x + 3)}^{3}}$

Step 5.8

Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

$\frac{300 (- 1 (2 x + 1))}{{(2 x + 3)}^{3}}$

Step 5.9

Move the negative in front of the fraction.

$- \frac{300 (2 x + 1)}{{(2 x + 3)}^{3}}$