Calculus Examples

$g (a - 3) = \frac{1}{(a + 3) - 3}$

Step 1

Remove parentheses.

$g (a - 3) = \frac{1}{a + 3 - 3}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d a} (g (a - 3)) = \frac{d}{d a} (\frac{1}{a + 3 - 3})$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

$g \frac{d}{d a} [a - 3] + (a - 3) \frac{d}{d a} [g]$

Step 3.2

Differentiate.

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Step 3.2.1

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

$g (\frac{d}{d a} [a] + \frac{d}{d a} [- 3]) + (a - 3) \frac{d}{d a} [g]$

Step 3.2.2

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

$g (1 + \frac{d}{d a} [- 3]) + (a - 3) \frac{d}{d a} [g]$

Step 3.2.3

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

$g (1 + 0) + (a - 3) \frac{d}{d a} [g]$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

$g \cdot 1 + (a - 3) \frac{d}{d a} [g]$

Step 3.2.4.2

Multiply $g$ by $1$ .

$g + (a - 3) \frac{d}{d a} [g]$

Step 3.3

Rewrite $\frac{d}{d a} [g]$ as $g'$ .

$g + (a - 3) g'$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$g + a g' - 3 g'$

Step 3.4.2

Reorder terms.

$a g' + g - 3 g'$

Step 4

Differentiate the right side of the equation.

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Step 4.1

Simplify the expression.

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Step 4.1.1

Subtract $3$ from $3$ .

$\frac{d}{d a} [\frac{1}{a + 0}]$

Step 4.1.2

Add $a$ and $0$ .

$\frac{d}{d a} [\frac{1}{a}]$

Step 4.1.3

Rewrite $\frac{1}{a}$ as $a^{- 1}$ .

$\frac{d}{d a} [a^{- 1}]$

Step 4.2

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

$- a^{- 2}$

Step 4.3

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

$- \frac{1}{a^{2}}$

Step 5

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

$a g' + g - 3 g' = - \frac{1}{a^{2}}$

Step 6

Solve for $g'$ .

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Step 6.1

Subtract $g$ from both sides of the equation.

$a g' - 3 g' = - \frac{1}{a^{2}} - g$

Step 6.2

Factor $g'$ out of $a g' - 3 g'$ .

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Step 6.2.1

Factor $g'$ out of $a g'$ .

$g' a - 3 g' = - \frac{1}{a^{2}} - g$

Step 6.2.2

Factor $g'$ out of $- 3 g'$ .

$g' a + g' \cdot - 3 = - \frac{1}{a^{2}} - g$

Step 6.2.3

Factor $g'$ out of $g' a + g' \cdot - 3$ .

$g' (a - 3) = - \frac{1}{a^{2}} - g$

Step 6.3

Divide each term in $g' (a - 3) = - \frac{1}{a^{2}} - g$ by $a - 3$ and simplify.

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Step 6.3.1

Divide each term in $g' (a - 3) = - \frac{1}{a^{2}} - g$ by $a - 3$ .

$\frac{g' (a - 3)}{a - 3} = \frac{- \frac{1}{a^{2}}}{a - 3} + \frac{- g}{a - 3}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $a - 3$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{g' (a - 3)}{a - 3} = \frac{- \frac{1}{a^{2}}}{a - 3} + \frac{- g}{a - 3}$

Step 6.3.2.1.2

Divide $g'$ by $1$ .

$g' = \frac{- \frac{1}{a^{2}}}{a - 3} + \frac{- g}{a - 3}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify each term.

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Step 6.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$g' = - \frac{1}{a^{2}} \cdot \frac{1}{a - 3} + \frac{- g}{a - 3}$

Step 6.3.3.1.2

Multiply $\frac{1}{a - 3}$ by $\frac{1}{a^{2}}$ .

$g' = - \frac{1}{(a - 3) a^{2}} + \frac{- g}{a - 3}$

Step 6.3.3.1.3

Move the negative in front of the fraction.

$g' = - \frac{1}{(a - 3) a^{2}} - \frac{g}{a - 3}$

Step 6.3.3.2

To write $- \frac{g}{a - 3}$ as a fraction with a common denominator, multiply by $\frac{a^{2}}{a^{2}}$ .

$g' = - \frac{1}{(a - 3) a^{2}} - \frac{g}{a - 3} \cdot \frac{a^{2}}{a^{2}}$

Step 6.3.3.3

Multiply $\frac{g}{a - 3}$ by $\frac{a^{2}}{a^{2}}$ .

$g' = - \frac{1}{(a - 3) a^{2}} - \frac{g a^{2}}{(a - 3) a^{2}}$

Step 6.3.3.4

Combine the numerators over the common denominator.

$g' = \frac{- 1 - g a^{2}}{(a - 3) a^{2}}$

Step 6.3.3.5

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

$g' = \frac{- 1 (1) - g a^{2}}{(a - 3) a^{2}}$

Step 6.3.3.6

Factor $- 1$ out of $- g a^{2}$ .

$g' = \frac{- 1 (1) - (g a^{2})}{(a - 3) a^{2}}$

Step 6.3.3.7

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

$g' = \frac{- 1 (1 + g a^{2})}{(a - 3) a^{2}}$

Step 6.3.3.8

Simplify the expression.

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Step 6.3.3.8.1

Move the negative in front of the fraction.

$g' = - \frac{1 + g a^{2}}{(a - 3) a^{2}}$

Step 6.3.3.8.2

Reorder factors in $- \frac{1 + g a^{2}}{(a - 3) a^{2}}$ .

$g' = - \frac{1 + g a^{2}}{a^{2} (a - 3)}$

Step 7

Replace $g'$ with $\frac{d g}{d a}$ .

$\frac{d g}{d a} = - \frac{1 + g a^{2}}{a^{2} (a - 3)}$