Calculus Examples

$y = 5 x^{- 2} + 9 x^{- 1}$

Step 1

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

$\frac{d}{d y} [5 \frac{1}{x^{2}} + 9 x^{- 1}]$

Step 2

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

$\frac{d}{d y} [5 \frac{1}{x^{2}} + 9 \frac{1}{x}]$

Step 3

By the Sum Rule, the derivative of $5 \frac{1}{x^{2}} + 9 \frac{1}{x}$ with respect to $y$ is $\frac{d}{d y} [5 \frac{1}{x^{2}}] + \frac{d}{d y} [9 \frac{1}{x}]$ .

$\frac{d}{d y} [5 \frac{1}{x^{2}}] + \frac{d}{d y} [9 \frac{1}{x}]$

Step 4

Evaluate $\frac{d}{d y} [5 \frac{1}{x^{2}}]$ .

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Step 4.1

Combine $5$ and $\frac{1}{x^{2}}$ .

$\frac{d}{d y} [\frac{5}{x^{2}}] + \frac{d}{d y} [9 \frac{1}{x}]$

Step 4.2

Since $\frac{5}{x^{2}}$ is constant with respect to $y$ , the derivative of $\frac{5}{x^{2}}$ with respect to $y$ is $0$ .

$0 + \frac{d}{d y} [9 \frac{1}{x}]$

Step 5

Evaluate $\frac{d}{d y} [9 \frac{1}{x}]$ .

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Step 5.1

Combine $9$ and $\frac{1}{x}$ .

$0 + \frac{d}{d y} [\frac{9}{x}]$

Step 5.2

Since $\frac{9}{x}$ is constant with respect to $y$ , the derivative of $\frac{9}{x}$ with respect to $y$ is $0$ .

$0 + 0$

Step 6

Add $0$ and $0$ .

$0$