Algebra Examples

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

Step 1

Differentiate.

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Step 1.1

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

$\frac{d}{d a} [a] + \frac{d}{d a} [\frac{1}{a^{2}}]$

Step 1.2

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

$1 + \frac{d}{d a} [\frac{1}{a^{2}}]$

Step 2

Evaluate $\frac{d}{d a} [\frac{1}{a^{2}}]$ .

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Step 2.1

Rewrite $\frac{1}{a^{2}}$ as ${(a^{2})}^{- 1}$ .

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

Step 2.2

Differentiate using the chain rule, which states that $\frac{d}{d a} [f (g (a))]$ is $f' (g (a)) g' (a)$ where $f (a) = a^{- 1}$ and $g (a) = a^{2}$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $a^{2}$ .

$1 + \frac{d}{d u} [u^{- 1}] \frac{d}{d a} [a^{2}]$

Step 2.2.2

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

$1 - u^{- 2} \frac{d}{d a} [a^{2}]$

Step 2.2.3

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

$1 - {(a^{2})}^{- 2} \frac{d}{d a} [a^{2}]$

Step 2.3

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

$1 - {(a^{2})}^{- 2} (2 a)$

Step 2.4

Multiply the exponents in ${(a^{2})}^{- 2}$ .

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Step 2.4.1

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

$1 - a^{2 \cdot - 2} (2 a)$

Step 2.4.2

Multiply $2$ by $- 2$ .

$1 - a^{- 4} (2 a)$

Step 2.5

Multiply $2$ by $- 1$ .

$1 - 2 a^{- 4} a$

Step 2.6

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

$1 - 2 (a^{1} a^{- 4})$

Step 2.7

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

$1 - 2 a^{1 - 4}$

Step 2.8

Subtract $4$ from $1$ .

$1 - 2 a^{- 3}$

Step 3

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

$1 - 2 \frac{1}{a^{3}}$

Step 4

Simplify.

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Step 4.1

Combine terms.

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Step 4.1.1

Combine $- 2$ and $\frac{1}{a^{3}}$ .

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

Step 4.1.2

Move the negative in front of the fraction.

$1 - \frac{2}{a^{3}}$

Step 4.2

Reorder terms.

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