Calculus Examples

$y = (4 t - 1) {(2 t - 2)}^{- 1}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d t} (y) = \frac{d}{d t} ((4 t - 1) {(2 t - 2)}^{- 1})$

Step 2

The derivative of $y$ with respect to $t$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{- 1}$ and $g (t) = 2 t - 2$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $2 t - 2$ .

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

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $2 t - 2$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $2$ by $1$ .

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

Step 3.3.5

Since $- 2$ is constant with respect to $t$ , the derivative of $- 2$ with respect to $t$ is $0$ .

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

Step 3.3.6

Simplify the expression.

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Step 3.3.6.1

Add $2$ and $0$ .

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

Step 3.3.6.2

Multiply $2$ by $- 1$ .

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

Step 3.3.7

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

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

Step 3.3.8

Since $4$ is constant with respect to $t$ , the derivative of $4 t$ with respect to $t$ is $4 \frac{d}{d t} [t]$ .

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

Step 3.3.9

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

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

Step 3.3.10

Multiply $4$ by $1$ .

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

Step 3.3.11

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

$(4 t - 1) (- 2 {(2 t - 2)}^{- 2}) + {(2 t - 2)}^{- 1} (4 + 0)$

Step 3.3.12

Simplify the expression.

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Step 3.3.12.1

Add $4$ and $0$ .

$(4 t - 1) (- 2 {(2 t - 2)}^{- 2}) + {(2 t - 2)}^{- 1} \cdot 4$

Step 3.3.12.2

Move $4$ to the left of ${(2 t - 2)}^{- 1}$ .

$(4 t - 1) (- 2 {(2 t - 2)}^{- 2}) + 4 \cdot {(2 t - 2)}^{- 1}$

Step 3.4

Simplify.

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Step 3.4.1

Reorder terms.

$- 2 {(2 t - 2)}^{- 2} (4 t - 1) + 4 {(2 t - 2)}^{- 1}$

Step 3.4.2

Simplify each term.

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Step 3.4.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.4.2.2

Simplify the denominator.

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Step 3.4.2.2.1

Factor $2$ out of $2 t - 2$ .

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Step 3.4.2.2.1.1

Factor $2$ out of $2 t$ .

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

Step 3.4.2.2.1.2

Factor $2$ out of $- 2$ .

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

Step 3.4.2.2.1.3

Factor $2$ out of $2 (t) + 2 (- 1)$ .

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

Step 3.4.2.2.2

Apply the product rule to $2 (t - 1)$ .

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

Step 3.4.2.2.3

Raise $2$ to the power of $2$ .

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

Step 3.4.2.3

Cancel the common factor of $2$ .

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Step 3.4.2.3.1

Factor $2$ out of $- 2$ .

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

Step 3.4.2.3.2

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

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

Step 3.4.2.3.3

Cancel the common factor.

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

Step 3.4.2.3.4

Rewrite the expression.

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

Step 3.4.2.4

Apply the distributive property.

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

Step 3.4.2.5

Cancel the common factor of $2$ .

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Step 3.4.2.5.1

Move the leading negative in $- \frac{1}{2 {(t - 1)}^{2}}$ into the numerator.

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

Step 3.4.2.5.2

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

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

Step 3.4.2.5.3

Factor $2$ out of $4 t$ .

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

Step 3.4.2.5.4

Cancel the common factor.

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

Step 3.4.2.5.5

Rewrite the expression.

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

Step 3.4.2.6

Combine $2$ and $\frac{- 1}{{(t - 1)}^{2}}$ .

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

Step 3.4.2.7

Multiply $2$ by $- 1$ .

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

Step 3.4.2.8

Combine $\frac{- 2}{{(t - 1)}^{2}}$ and $t$ .

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

Step 3.4.2.9

Multiply $- \frac{1}{2 {(t - 1)}^{2}} \cdot - 1$ .

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Step 3.4.2.9.1

Multiply $- 1$ by $- 1$ .

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

Step 3.4.2.9.2

Multiply $\frac{1}{2 {(t - 1)}^{2}}$ by $1$ .

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

Step 3.4.2.10

Move the negative in front of the fraction.

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

Step 3.4.2.11

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.4.2.12

Factor $2$ out of $2 t - 2$ .

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Step 3.4.2.12.1

Factor $2$ out of $2 t$ .

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

Step 3.4.2.12.2

Factor $2$ out of $- 2$ .

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

Step 3.4.2.12.3

Factor $2$ out of $2 (t) + 2 (- 1)$ .

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

Step 3.4.2.13

Cancel the common factor of $2$ .

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Step 3.4.2.13.1

Factor $2$ out of $4$ .

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

Step 3.4.2.13.2

Cancel the common factor.

$- \frac{2 t}{{(t - 1)}^{2}} + \frac{1}{2 {(t - 1)}^{2}} + 2 \cdot 2 \frac{1}{2 (t - 1)}$

Step 3.4.2.13.3

Rewrite the expression.

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

Step 3.4.2.14

Combine $2$ and $\frac{1}{t - 1}$ .

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

Step 3.4.3

To write $\frac{2}{t - 1}$ as a fraction with a common denominator, multiply by $\frac{t - 1}{t - 1}$ .

$- \frac{2 t}{{(t - 1)}^{2}} + \frac{2}{t - 1} \cdot \frac{t - 1}{t - 1} + \frac{1}{2 {(t - 1)}^{2}}$

Step 3.4.4

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

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Step 3.4.4.1

Multiply $\frac{2}{t - 1}$ by $\frac{t - 1}{t - 1}$ .

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

Step 3.4.4.2

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

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

Step 3.4.4.3

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

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

Step 3.4.4.4

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

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

Step 3.4.4.5

Add $1$ and $1$ .

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

Step 3.4.5

Combine the numerators over the common denominator.

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

Step 3.4.6

Simplify each term.

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Step 3.4.6.1

Simplify the numerator.

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Step 3.4.6.1.1

Factor $2$ out of $- 2 t + 2 (t - 1)$ .

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Step 3.4.6.1.1.1

Factor $2$ out of $- 2 t$ .

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

Step 3.4.6.1.1.2

Factor $2$ out of $2 (- t) + 2 (t - 1)$ .

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

Step 3.4.6.1.2

Add $- t$ and $t$ .

$\frac{2 (0 - 1)}{{(t - 1)}^{2}} + \frac{1}{2 {(t - 1)}^{2}}$

Step 3.4.6.1.3

Subtract $1$ from $0$ .

$\frac{2 \cdot - 1}{{(t - 1)}^{2}} + \frac{1}{2 {(t - 1)}^{2}}$

Step 3.4.6.2

Multiply $2$ by $- 1$ .

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

Step 3.4.6.3

Move the negative in front of the fraction.

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

Step 3.4.7

To write $- \frac{2}{{(t - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{2}{{(t - 1)}^{2}} \cdot \frac{2}{2} + \frac{1}{2 {(t - 1)}^{2}}$

Step 3.4.8

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

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Step 3.4.8.1

Multiply $\frac{2}{{(t - 1)}^{2}}$ by $\frac{2}{2}$ .

$- \frac{2 \cdot 2}{{(t - 1)}^{2} \cdot 2} + \frac{1}{2 {(t - 1)}^{2}}$

Step 3.4.8.2

Reorder the factors of ${(t - 1)}^{2} \cdot 2$ .

$- \frac{2 \cdot 2}{2 {(t - 1)}^{2}} + \frac{1}{2 {(t - 1)}^{2}}$

Step 3.4.9

Combine the numerators over the common denominator.

$\frac{- 2 \cdot 2 + 1}{2 {(t - 1)}^{2}}$

Step 3.4.10

Simplify the numerator.

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Step 3.4.10.1

Multiply $- 2$ by $2$ .

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

Step 3.4.10.2

Add $- 4$ and $1$ .

$\frac{- 3}{2 {(t - 1)}^{2}}$

Step 3.4.11

Move the negative in front of the fraction.

$- \frac{3}{2 {(t - 1)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{3}{2 {(t - 1)}^{2}}$

Step 5

Replace $y'$ with $\frac{d y}{d t}$ .

$\frac{d y}{d t} = - \frac{3}{2 {(t - 1)}^{2}}$