Calculus Examples

$y = \frac{t^{2} + 3}{t^{4} - 5 t^{2} + 1}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t^{2} + 3$ and $g (t) = t^{4} - 5 t^{2} + 1$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $2 t$ and $0$ .

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

Step 2.4.2

Move $2$ to the left of $t^{4} - 5 t^{2} + 1$ .

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

Step 2.5

By the Sum Rule, the derivative of $t^{4} - 5 t^{2} + 1$ with respect to $t$ is $\frac{d}{d t} [t^{4}] + \frac{d}{d t} [- 5 t^{2}] + \frac{d}{d t} [1]$ .

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

Step 2.6

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 4$ .

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

Step 2.7

Since $- 5$ is constant with respect to $t$ , the derivative of $- 5 t^{2}$ with respect to $t$ is $- 5 \frac{d}{d t} [t^{2}]$ .

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

Step 2.8

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

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

Step 2.9

Multiply $2$ by $- 5$ .

$\frac{2 (t^{4} - 5 t^{2} + 1) t - (t^{2} + 3) (4 t^{3} - 10 t + \frac{d}{d t} [1])}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 2.10

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

$\frac{2 (t^{4} - 5 t^{2} + 1) t - (t^{2} + 3) (4 t^{3} - 10 t + 0)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 2.11

Add $4 t^{3} - 10 t$ and $0$ .

$\frac{2 (t^{4} - 5 t^{2} + 1) t - (t^{2} + 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(2 t^{4} + 2 (- 5 t^{2}) + 2 \cdot 1) t - (t^{2} + 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.2

Apply the distributive property.

$\frac{2 t^{4} t + 2 (- 5 t^{2}) t + 2 \cdot 1 t - (t^{2} + 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.3

Apply the distributive property.

$\frac{2 t^{4} t + 2 (- 5 t^{2}) t + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4

Simplify the numerator.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Multiply $t^{4}$ by $t$ by adding the exponents.

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Step 3.4.1.1.1

Move $t$ .

$\frac{2 (t \cdot t^{4}) + 2 (- 5 t^{2}) t + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.1.2

Multiply $t$ by $t^{4}$ .

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Step 3.4.1.1.2.1

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

$\frac{2 (t^{1} t^{4}) + 2 (- 5 t^{2}) t + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.1.2.2

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

$\frac{2 t^{1 + 4} + 2 (- 5 t^{2}) t + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.1.3

Add $1$ and $4$ .

$\frac{2 t^{5} + 2 (- 5 t^{2}) t + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.2

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

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Step 3.4.1.2.1

Move $t$ .

$\frac{2 t^{5} + 2 (- 5 (t \cdot t^{2})) + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.2.2

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

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Step 3.4.1.2.2.1

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

$\frac{2 t^{5} + 2 (- 5 (t^{1} t^{2})) + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.2.2.2

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

$\frac{2 t^{5} + 2 (- 5 t^{1 + 2}) + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.2.3

Add $1$ and $2$ .

$\frac{2 t^{5} + 2 (- 5 t^{3}) + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.3

Multiply $- 5$ by $2$ .

$\frac{2 t^{5} - 10 t^{3} + 2 \cdot 1 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.4

Multiply $2$ by $1$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t + (- t^{2} - 1 \cdot 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.5

Multiply $- 1$ by $3$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t + (- t^{2} - 3) (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.6

Expand $(- t^{2} - 3) (4 t^{3} - 10 t)$ using the FOIL Method.

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Step 3.4.1.6.1

Apply the distributive property.

$\frac{2 t^{5} - 10 t^{3} + 2 t - t^{2} (4 t^{3} - 10 t) - 3 (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.6.2

Apply the distributive property.

$\frac{2 t^{5} - 10 t^{3} + 2 t - t^{2} (4 t^{3}) - t^{2} (- 10 t) - 3 (4 t^{3} - 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.6.3

Apply the distributive property.

$\frac{2 t^{5} - 10 t^{3} + 2 t - t^{2} (4 t^{3}) - t^{2} (- 10 t) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7

Simplify and combine like terms.

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Step 3.4.1.7.1

Simplify each term.

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Step 3.4.1.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 t^{5} - 10 t^{3} + 2 t - 1 \cdot 4 t^{2} t^{3} - t^{2} (- 10 t) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.2

Multiply $t^{2}$ by $t^{3}$ by adding the exponents.

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Step 3.4.1.7.1.2.1

Move $t^{3}$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 1 \cdot 4 (t^{3} t^{2}) - t^{2} (- 10 t) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.2.2

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

$\frac{2 t^{5} - 10 t^{3} + 2 t - 1 \cdot 4 t^{3 + 2} - t^{2} (- 10 t) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.2.3

Add $3$ and $2$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 1 \cdot 4 t^{5} - t^{2} (- 10 t) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.3

Multiply $- 1$ by $4$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} - t^{2} (- 10 t) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.4

Rewrite using the commutative property of multiplication.

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} - 1 \cdot - 10 t^{2} t - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.5

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

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Step 3.4.1.7.1.5.1

Move $t$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} - 1 \cdot - 10 (t \cdot t^{2}) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.5.2

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

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Step 3.4.1.7.1.5.2.1

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

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} - 1 \cdot - 10 (t^{1} t^{2}) - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.5.2.2

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

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} - 1 \cdot - 10 t^{1 + 2} - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.5.3

Add $1$ and $2$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} - 1 \cdot - 10 t^{3} - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.6

Multiply $- 1$ by $- 10$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} + 10 t^{3} - 3 (4 t^{3}) - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.7

Multiply $4$ by $- 3$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} + 10 t^{3} - 12 t^{3} - 3 (- 10 t)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.1.8

Multiply $- 10$ by $- 3$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} + 10 t^{3} - 12 t^{3} + 30 t}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.1.7.2

Subtract $12 t^{3}$ from $10 t^{3}$ .

$\frac{2 t^{5} - 10 t^{3} + 2 t - 4 t^{5} - 2 t^{3} + 30 t}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.2

Subtract $4 t^{5}$ from $2 t^{5}$ .

$\frac{- 2 t^{5} - 10 t^{3} + 2 t - 2 t^{3} + 30 t}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.3

Subtract $2 t^{3}$ from $- 10 t^{3}$ .

$\frac{- 2 t^{5} - 12 t^{3} + 2 t + 30 t}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.4.4

Add $2 t$ and $30 t$ .

$\frac{- 2 t^{5} - 12 t^{3} + 32 t}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $2 t$ out of $- 2 t^{5} - 12 t^{3} + 32 t$ .

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Step 3.5.1.1

Factor $2 t$ out of $- 2 t^{5}$ .

$\frac{2 t (- t^{4}) - 12 t^{3} + 32 t}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.1.2

Factor $2 t$ out of $- 12 t^{3}$ .

$\frac{2 t (- t^{4}) + 2 t (- 6 t^{2}) + 32 t}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.1.3

Factor $2 t$ out of $32 t$ .

$\frac{2 t (- t^{4}) + 2 t (- 6 t^{2}) + 2 t (16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.1.4

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

$\frac{2 t (- t^{4} - 6 t^{2}) + 2 t (16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.1.5

Factor $2 t$ out of $2 t (- t^{4} - 6 t^{2}) + 2 t (16)$ .

$\frac{2 t (- t^{4} - 6 t^{2} + 16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.2

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$\frac{2 t (- {(t^{2})}^{2} - 6 t^{2} + 16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.3

Let $u = t^{2}$ . Substitute $u$ for all occurrences of $t^{2}$ .

$\frac{2 t (- u^{2} - 6 u + 16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.4

Factor by grouping.

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Step 3.5.4.1

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 $a \cdot c = - 1 \cdot 16 = - 16$ and whose sum is $b = - 6$ .

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Step 3.5.4.1.1

Factor $- 6$ out of $- 6 u$ .

$\frac{2 t (- u^{2} - 6 (u) + 16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.4.1.2

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

$\frac{2 t (- u^{2} + (2 - 8) u + 16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.4.1.3

Apply the distributive property.

$\frac{2 t (- u^{2} + 2 u - 8 u + 16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.4.2

Factor out the greatest common factor from each group.

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Step 3.5.4.2.1

Group the first two terms and the last two terms.

$\frac{2 t ((- u^{2} + 2 u) - 8 u + 16)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.4.2.2

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

$\frac{2 t (u (- u + 2) + 8 (- u + 2))}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.4.3

Factor the polynomial by factoring out the greatest common factor, $- u + 2$ .

$\frac{2 t ((- u + 2) (u + 8))}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.5.5

Replace all occurrences of $u$ with $t^{2}$ .

$\frac{2 t (- t^{2} + 2) (t^{2} + 8)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.6

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

$\frac{2 t (- (t^{2}) + 2) (t^{2} + 8)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.7

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

$\frac{2 t (- (t^{2}) - 1 (- 2)) (t^{2} + 8)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.8

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

$\frac{2 t (- (t^{2} - 2)) (t^{2} + 8)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.9

Rewrite $- (t^{2} - 2)$ as $- 1 (t^{2} - 2)$ .

$\frac{2 t (- 1 (t^{2} - 2)) (t^{2} + 8)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{((2 t) (t^{2} - 2)) (t^{2} + 8)}{{(t^{4} - 5 t^{2} + 1)}^{2}}$