Precalculus Examples

$P (t) = \frac{600 t}{3 t^{2} + 4}$

Step 1

Consider the difference quotient formula.

$\frac{f (t + h) - f t}{h}$

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at $x = t + h$ .

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Step 2.1.1

Replace the variable $t$ with $t + h$ in the expression.

$P (t + h) = \frac{600 (t + h)}{3 {(t + h)}^{2} + 4}$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the denominator.

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Step 2.1.2.1.1

Rewrite ${(t + h)}^{2}$ as $(t + h) (t + h)$ .

$P (t + h) = \frac{600 (t + h)}{3 ((t + h) (t + h)) + 4}$

Step 2.1.2.1.2

Expand $(t + h) (t + h)$ using the FOIL Method.

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Step 2.1.2.1.2.1

Apply the distributive property.

$P (t + h) = \frac{600 (t + h)}{3 (t (t + h) + h (t + h)) + 4}$

Step 2.1.2.1.2.2

Apply the distributive property.

$P (t + h) = \frac{600 (t + h)}{3 (t \cdot t + t h + h (t + h)) + 4}$

Step 2.1.2.1.2.3

Apply the distributive property.

$P (t + h) = \frac{600 (t + h)}{3 (t \cdot t + t h + h t + h \cdot h) + 4}$

Step 2.1.2.1.3

Simplify and combine like terms.

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Step 2.1.2.1.3.1

Simplify each term.

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Step 2.1.2.1.3.1.1

Multiply $t$ by $t$ .

$P (t + h) = \frac{600 (t + h)}{3 (t^{2} + t h + h t + h \cdot h) + 4}$

Step 2.1.2.1.3.1.2

Multiply $h$ by $h$ .

$P (t + h) = \frac{600 (t + h)}{3 (t^{2} + t h + h t + h^{2}) + 4}$

Step 2.1.2.1.3.2

Add $t h$ and $h t$ .

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Step 2.1.2.1.3.2.1

Reorder $t$ and $h$ .

$P (t + h) = \frac{600 (t + h)}{3 (t^{2} + h t + h t + h^{2}) + 4}$

Step 2.1.2.1.3.2.2

Add $h t$ and $h t$ .

$P (t + h) = \frac{600 (t + h)}{3 (t^{2} + 2 h t + h^{2}) + 4}$

Step 2.1.2.1.4

Apply the distributive property.

$P (t + h) = \frac{600 (t + h)}{3 t^{2} + 3 (2 h t) + 3 h^{2} + 4}$

Step 2.1.2.1.5

Multiply $2$ by $3$ .

$P (t + h) = \frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4}$

Step 2.1.2.2

The final answer is $\frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4}$ .

$\frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4}$

Step 2.2

Find the components of the definition.

$P (t + h) = \frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4}$

$P (t) = \frac{600 t}{3 t^{2} + 4}$

$P (t + h) = \frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4}$

$P (t) = \frac{600 t}{3 t^{2} + 4}$

Step 3

Plug in the components.

$\frac{P (t + h) - P t}{h} = \frac{\frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4} - (\frac{600 t}{3 t^{2} + 4})}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4}$ as a fraction with a common denominator, multiply by $\frac{3 t^{2} + 4}{3 t^{2} + 4}$ .

$\frac{\frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4} \cdot \frac{3 t^{2} + 4}{3 t^{2} + 4} - \frac{600 t}{3 t^{2} + 4}}{h}$

Step 4.1.2

To write $- \frac{600 t}{3 t^{2} + 4}$ as a fraction with a common denominator, multiply by $\frac{3 t^{2} + 6 h t + 3 h^{2} + 4}{3 t^{2} + 6 h t + 3 h^{2} + 4}$ .

$\frac{\frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4} \cdot \frac{3 t^{2} + 4}{3 t^{2} + 4} - \frac{600 t}{3 t^{2} + 4} \cdot \frac{3 t^{2} + 6 h t + 3 h^{2} + 4}{3 t^{2} + 6 h t + 3 h^{2} + 4}}{h}$

Step 4.1.3

Write each expression with a common denominator of $(3 t^{2} + 6 h t + 3 h^{2} + 4) (3 t^{2} + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{600 (t + h)}{3 t^{2} + 6 h t + 3 h^{2} + 4}$ by $\frac{3 t^{2} + 4}{3 t^{2} + 4}$ .

$\frac{\frac{600 (t + h) (3 t^{2} + 4)}{(3 t^{2} + 6 h t + 3 h^{2} + 4) (3 t^{2} + 4)} - \frac{600 t}{3 t^{2} + 4} \cdot \frac{3 t^{2} + 6 h t + 3 h^{2} + 4}{3 t^{2} + 6 h t + 3 h^{2} + 4}}{h}$

Step 4.1.3.2

Multiply $\frac{600 t}{3 t^{2} + 4}$ by $\frac{3 t^{2} + 6 h t + 3 h^{2} + 4}{3 t^{2} + 6 h t + 3 h^{2} + 4}$ .

$\frac{\frac{600 (t + h) (3 t^{2} + 4)}{(3 t^{2} + 6 h t + 3 h^{2} + 4) (3 t^{2} + 4)} - \frac{600 t (3 t^{2} + 6 h t + 3 h^{2} + 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.3.3

Reorder the factors of $(3 t^{2} + 6 h t + 3 h^{2} + 4) (3 t^{2} + 4)$ .

$\frac{\frac{600 (t + h) (3 t^{2} + 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)} - \frac{600 t (3 t^{2} + 6 h t + 3 h^{2} + 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.4

Combine the numerators over the common denominator.

$\frac{\frac{600 (t + h) (3 t^{2} + 4) - 600 t (3 t^{2} + 6 h t + 3 h^{2} + 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5

Rewrite $\frac{600 (t + h) (3 t^{2} + 4) - 600 t (3 t^{2} + 6 h t + 3 h^{2} + 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$ in a factored form.

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Step 4.1.5.1

Factor $600$ out of $600 (t + h) (3 t^{2} + 4) - 600 t (3 t^{2} + 6 h t + 3 h^{2} + 4)$ .

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Step 4.1.5.1.1

Factor $600$ out of $600 (t + h) (3 t^{2} + 4)$ .

$\frac{\frac{600 ((t + h) (3 t^{2} + 4)) - 600 t (3 t^{2} + 6 h t + 3 h^{2} + 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.1.2

Factor $600$ out of $- 600 t (3 t^{2} + 6 h t + 3 h^{2} + 4)$ .

$\frac{\frac{600 ((t + h) (3 t^{2} + 4)) + 600 (- t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.1.3

Factor $600$ out of $600 ((t + h) (3 t^{2} + 4)) + 600 (- t (3 t^{2} + 6 h t + 3 h^{2} + 4))$ .

$\frac{\frac{600 ((t + h) (3 t^{2} + 4) - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.2

Expand $(t + h) (3 t^{2} + 4)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

$\frac{\frac{600 (t (3 t^{2} + 4) + h (3 t^{2} + 4) - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.2.2

Apply the distributive property.

$\frac{\frac{600 (t (3 t^{2}) + t \cdot 4 + h (3 t^{2} + 4) - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.2.3

Apply the distributive property.

$\frac{\frac{600 (t (3 t^{2}) + t \cdot 4 + h (3 t^{2}) + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3

Simplify each term.

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Step 4.1.5.3.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{600 (3 t \cdot t^{2} + t \cdot 4 + h (3 t^{2}) + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3.2

Multiply $t$ by $t^{2}$ by adding the exponents.

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Step 4.1.5.3.2.1

Move $t^{2}$ .

$\frac{\frac{600 (3 (t^{2} t) + t \cdot 4 + h (3 t^{2}) + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3.2.2

Multiply $t^{2}$ by $t$ .

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Step 4.1.5.3.2.2.1

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

$\frac{\frac{600 (3 (t^{2} t^{1}) + t \cdot 4 + h (3 t^{2}) + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3.2.2.2

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

$\frac{\frac{600 (3 t^{2 + 1} + t \cdot 4 + h (3 t^{2}) + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3.2.3

Add $2$ and $1$ .

$\frac{\frac{600 (3 t^{3} + t \cdot 4 + h (3 t^{2}) + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3.3

Move $4$ to the left of $t$ .

$\frac{\frac{600 (3 t^{3} + 4 \cdot t + h (3 t^{2}) + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3.4

Rewrite using the commutative property of multiplication.

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + h \cdot 4 - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.3.5

Move $4$ to the left of $h$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - t (3 t^{2} + 6 h t + 3 h^{2} + 4))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.4

Apply the distributive property.

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - t (3 t^{2}) - t (6 h t) - t (3 h^{2}) - t \cdot 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.5

Simplify.

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Step 4.1.5.5.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 t \cdot t^{2} - t (6 h t) - t (3 h^{2}) - t \cdot 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.5.2

Multiply $t$ by $t$ by adding the exponents.

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Step 4.1.5.5.2.1

Move $t$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 t \cdot t^{2} - (t \cdot t) (6 h) - t (3 h^{2}) - t \cdot 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.5.2.2

Multiply $t$ by $t$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 t \cdot t^{2} - t^{2} (6 h) - t (3 h^{2}) - t \cdot 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.5.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 t \cdot t^{2} - t^{2} (6 h) - 1 \cdot 3 t h^{2} - t \cdot 4)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.5.4

Multiply $4$ by $- 1$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 t \cdot t^{2} - t^{2} (6 h) - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6

Simplify each term.

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Step 4.1.5.6.1

Multiply $t$ by $t^{2}$ by adding the exponents.

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Step 4.1.5.6.1.1

Move $t^{2}$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 (t^{2} t) - t^{2} (6 h) - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6.1.2

Multiply $t^{2}$ by $t$ .

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Step 4.1.5.6.1.2.1

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

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 (t^{2} t^{1}) - t^{2} (6 h) - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6.1.2.2

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

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 t^{2 + 1} - t^{2} (6 h) - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6.1.3

Add $2$ and $1$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 1 \cdot 3 t^{3} - t^{2} (6 h) - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6.2

Multiply $- 1$ by $3$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 3 t^{3} - t^{2} (6 h) - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6.3

Rewrite using the commutative property of multiplication.

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 3 t^{3} - 1 \cdot 6 t^{2} h - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6.4

Multiply $- 1$ by $6$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 3 t^{3} - 6 t^{2} h - 1 \cdot 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.6.5

Multiply $- 1$ by $3$ .

$\frac{\frac{600 (3 t^{3} + 4 t + 3 h t^{2} + 4 h - 3 t^{3} - 6 t^{2} h - 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.7

Subtract $3 t^{3}$ from $3 t^{3}$ .

$\frac{\frac{600 (4 t + 3 h t^{2} + 4 h + 0 - 6 t^{2} h - 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.8

Add $4 t$ and $0$ .

$\frac{\frac{600 (4 t + 3 h t^{2} + 4 h - 6 t^{2} h - 3 t h^{2} - 4 t)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.9

Subtract $4 t$ from $4 t$ .

$\frac{\frac{600 (3 h t^{2} + 4 h - 6 t^{2} h - 3 t h^{2} + 0)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.10

Add $3 h t^{2} + 4 h - 6 t^{2} h - 3 t h^{2}$ and $0$ .

$\frac{\frac{600 (3 h t^{2} + 4 h - 6 t^{2} h - 3 t h^{2})}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.11

Subtract $6 t^{2} h$ from $3 h t^{2}$ .

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Step 4.1.5.11.1

Move $h$ .

$\frac{\frac{600 (3 t^{2} h - 6 t^{2} h + 4 h - 3 t h^{2})}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.11.2

Subtract $6 t^{2} h$ from $3 t^{2} h$ .

$\frac{\frac{600 (- 3 t^{2} h + 4 h - 3 t h^{2})}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.12

Factor $h$ out of $- 3 t^{2} h + 4 h - 3 t h^{2}$ .

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Step 4.1.5.12.1

Factor $h$ out of $- 3 t^{2} h$ .

$\frac{\frac{600 (h (- 3 t^{2}) + 4 h - 3 t h^{2})}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.12.2

Factor $h$ out of $4 h$ .

$\frac{\frac{600 (h (- 3 t^{2}) + h \cdot 4 - 3 t h^{2})}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.12.3

Factor $h$ out of $- 3 t h^{2}$ .

$\frac{\frac{600 (h (- 3 t^{2}) + h \cdot 4 + h (- 3 t h))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.12.4

Factor $h$ out of $h (- 3 t^{2}) + h \cdot 4$ .

$\frac{\frac{600 (h (- 3 t^{2} + 4) + h (- 3 t h))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.1.5.12.5

Factor $h$ out of $h (- 3 t^{2} + 4) + h (- 3 t h)$ .

$\frac{\frac{600 (h (- 3 t^{2} + 4 - 3 t h))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

$\frac{\frac{600 h (- 3 t^{2} + 4 - 3 t h)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{600 h (- 3 t^{2} + 4 - 3 t h)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)} \cdot \frac{1}{h}$

Step 4.3

Combine.

$\frac{600 h (- 3 t^{2} + 4 - 3 t h) \cdot 1}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4) h}$

Step 4.4

Cancel the common factor of $h$ .

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Step 4.4.1

Cancel the common factor.

$\frac{600 h (- 3 t^{2} + 4 - 3 t h) \cdot 1}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4) h}$

Step 4.4.2

Rewrite the expression.

$\frac{600 (- 3 t^{2} + 4 - 3 t h) \cdot 1}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.5

Multiply $600$ by $1$ .

$\frac{600 (- 3 t^{2} + 4 - 3 t h)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.6

Factor $- 1$ out of $- 3 t^{2}$ .

$\frac{600 (- (3 t^{2}) + 4 - 3 t h)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.7

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

$\frac{600 (- (3 t^{2}) - 1 (- 4) - 3 t h)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.8

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

$\frac{600 (- (3 t^{2} - 4) - 3 t h)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.9

Factor $- 1$ out of $- 3 t h$ .

$\frac{600 (- (3 t^{2} - 4) - (3 t h))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.10

Factor $- 1$ out of $- (3 t^{2} - 4) - (3 t h)$ .

$\frac{600 (- (3 t^{2} - 4 + 3 t h))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.11

Simplify the expression.

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Step 4.11.1

Rewrite $- (3 t^{2} - 4 + 3 t h)$ as $- 1 (3 t^{2} - 4 + 3 t h)$ .

$\frac{600 (- 1 (3 t^{2} - 4 + 3 t h))}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 4.11.2

Move the negative in front of the fraction.

$- \frac{600 (3 t^{2} - 4 + 3 t h)}{(3 t^{2} + 4) (3 t^{2} + 6 h t + 3 h^{2} + 4)}$

Step 5