Calculus Examples

$A = 2 x^{2} + \frac{2916}{x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (A) = \frac{d}{d x} (2 x^{2} + \frac{2916}{x})$

Step 2

The derivative of $A$ with respect to $x$ is $A'$ .

$A'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $2 x^{2} + \frac{2916}{x}$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [\frac{2916}{x}]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [\frac{2916}{x}]$

Step 3.2

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

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Step 3.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{2916}{x}]$

Step 3.2.2

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

$2 (2 x) + \frac{d}{d x} [\frac{2916}{x}]$

Step 3.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [\frac{2916}{x}]$

Step 3.3

Evaluate $\frac{d}{d x} [\frac{2916}{x}]$ .

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Step 3.3.1

Since $2916$ is constant with respect to $x$ , the derivative of $\frac{2916}{x}$ with respect to $x$ is $2916 \frac{d}{d x} [\frac{1}{x}]$ .

$4 x + 2916 \frac{d}{d x} [\frac{1}{x}]$

Step 3.3.2

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

$4 x + 2916 \frac{d}{d x} [x^{- 1}]$

Step 3.3.3

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

$4 x + 2916 (- x^{- 2})$

Step 3.3.4

Multiply $- 1$ by $2916$ .

$4 x - 2916 x^{- 2}$

Step 3.4

Simplify.

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Step 3.4.1

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

$4 x - 2916 \frac{1}{x^{2}}$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Combine $- 2916$ and $\frac{1}{x^{2}}$ .

$4 x + \frac{- 2916}{x^{2}}$

Step 3.4.2.2

Move the negative in front of the fraction.

$4 x - \frac{2916}{x^{2}}$

Step 4

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

$A' = 4 x - \frac{2916}{x^{2}}$

Step 5

Replace $A'$ with $\frac{d A}{d x}$ .

$\frac{d A}{d x} = 4 x - \frac{2916}{x^{2}}$