Calculus Examples

$y = 3 x^{2} - 11 x - 8 x^{- 3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3 x^{2} - 11 x - 8 x^{- 3})$

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

By the Sum Rule, the derivative of $3 x^{2} - 11 x - 8 x^{- 3}$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [- 8 x^{- 3}]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [- 8 x^{- 3}]$

Step 3.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 3.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [- 8 x^{- 3}]$

Step 3.2.2

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

$3 (2 x) + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [- 8 x^{- 3}]$

Step 3.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [- 11 x] + \frac{d}{d x} [- 8 x^{- 3}]$

Step 3.3

Evaluate $\frac{d}{d x} [- 11 x]$ .

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Step 3.3.1

Since $- 11$ is constant with respect to $x$ , the derivative of $- 11 x$ with respect to $x$ is $- 11 \frac{d}{d x} [x]$ .

$6 x - 11 \frac{d}{d x} [x] + \frac{d}{d x} [- 8 x^{- 3}]$

Step 3.3.2

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

$6 x - 11 \cdot 1 + \frac{d}{d x} [- 8 x^{- 3}]$

Step 3.3.3

Multiply $- 11$ by $1$ .

$6 x - 11 + \frac{d}{d x} [- 8 x^{- 3}]$

Step 3.4

Evaluate $\frac{d}{d x} [- 8 x^{- 3}]$ .

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Step 3.4.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 x^{- 3}$ with respect to $x$ is $- 8 \frac{d}{d x} [x^{- 3}]$ .

$6 x - 11 - 8 \frac{d}{d x} [x^{- 3}]$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$6 x - 11 - 8 (- 3 x^{- 4})$

Step 3.4.3

Multiply $- 3$ by $- 8$ .

$6 x - 11 + 24 x^{- 4}$

Step 3.5

Simplify.

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Step 3.5.1

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

$6 x - 11 + 24 \frac{1}{x^{4}}$

Step 3.5.2

Combine $24$ and $\frac{1}{x^{4}}$ .

$6 x - 11 + \frac{24}{x^{4}}$

Step 3.5.3

Reorder terms.

$6 x + \frac{24}{x^{4}} - 11$

Step 4

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

$y' = 6 x + \frac{24}{x^{4}} - 11$

Step 5

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

$\frac{d y}{d x} = 6 x + \frac{24}{x^{4}} - 11$