Calculus Examples

$f (x) = c x^{2} + 6 x^{- 2}$

Step 1

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

$\frac{d}{d c} [c x^{2} + 6 \frac{1}{x^{2}}]$

Step 2

By the Sum Rule, the derivative of $c x^{2} + 6 \frac{1}{x^{2}}$ with respect to $c$ is $\frac{d}{d c} [c x^{2}] + \frac{d}{d c} [6 \frac{1}{x^{2}}]$ .

$\frac{d}{d c} [c x^{2}] + \frac{d}{d c} [6 \frac{1}{x^{2}}]$

Step 3

Evaluate $\frac{d}{d c} [c x^{2}]$ .

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Step 3.1

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

$x^{2} \frac{d}{d c} [c] + \frac{d}{d c} [6 \frac{1}{x^{2}}]$

Step 3.2

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

$x^{2} \cdot 1 + \frac{d}{d c} [6 \frac{1}{x^{2}}]$

Step 3.3

Multiply $x^{2}$ by $1$ .

$x^{2} + \frac{d}{d c} [6 \frac{1}{x^{2}}]$

Step 4

Evaluate $\frac{d}{d c} [6 \frac{1}{x^{2}}]$ .

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Step 4.1

Combine $6$ and $\frac{1}{x^{2}}$ .

$x^{2} + \frac{d}{d c} [\frac{6}{x^{2}}]$

Step 4.2

Since $\frac{6}{x^{2}}$ is constant with respect to $c$ , the derivative of $\frac{6}{x^{2}}$ with respect to $c$ is $0$ .

$x^{2} + 0$

Step 5

Add $x^{2}$ and $0$ .

$x^{2}$