Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{- 1} + x^{- 2} + 2^{x})$

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

$- x^{- 2} - 2 x^{- 3} + \frac{d}{d x} [2^{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$- x^{- 2} - 2 x^{- 3} + 2^{x} \ln (2)$

Step 3.3

Simplify.

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Step 3.3.1

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

$- \frac{1}{x^{2}} - 2 x^{- 3} + 2^{x} \ln (2)$

Step 3.3.2

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

$- \frac{1}{x^{2}} - 2 \frac{1}{x^{3}} + 2^{x} \ln (2)$

Step 3.3.3

Combine terms.

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Step 3.3.3.1

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

$- \frac{1}{x^{2}} + \frac{- 2}{x^{3}} + 2^{x} \ln (2)$

Step 3.3.3.2

Move the negative in front of the fraction.

$- \frac{1}{x^{2}} - \frac{2}{x^{3}} + 2^{x} \ln (2)$

Step 3.3.4

Reorder terms.

$2^{x} \ln (2) - \frac{1}{x^{2}} - \frac{2}{x^{3}}$

Step 4

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

$y' = 2^{x} \ln (2) - \frac{1}{x^{2}} - \frac{2}{x^{3}}$

Step 5

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

$\frac{d y}{d x} = 2^{x} \ln (2) - \frac{1}{x^{2}} - \frac{2}{x^{3}}$