Calculus Examples

$y = x^{3} - \frac{x^{2}}{28} + 8 x + 1$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{3} - \frac{x^{2}}{28} + 8 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{x^{2}}{28}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{x^{2}}{28}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 1.2

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

$3 x^{2} + \frac{d}{d x} [- \frac{x^{2}}{28}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2

Evaluate $\frac{d}{d x} [- \frac{x^{2}}{28}]$ .

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Step 2.1

Since $- \frac{1}{28}$ is constant with respect to $x$ , the derivative of $- \frac{x^{2}}{28}$ with respect to $x$ is $- \frac{1}{28} \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - \frac{1}{28} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.2

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

$3 x^{2} - \frac{1}{28} (2 x) + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.3

Multiply $2$ by $- 1$ .

$3 x^{2} - 2 (\frac{1}{28}) x + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.4

Combine $- 2$ and $\frac{1}{28}$ .

$3 x^{2} + \frac{- 2}{28} x + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.5

Combine $\frac{- 2}{28}$ and $x$ .

$3 x^{2} + \frac{- 2 x}{28} + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.6

Cancel the common factor of $- 2$ and $28$ .

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Step 2.6.1

Factor $2$ out of $- 2 x$ .

$3 x^{2} + \frac{2 (- x)}{28} + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $28$ .

$3 x^{2} + \frac{2 (- x)}{2 (14)} + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.6.2.2

Cancel the common factor.

$3 x^{2} + \frac{2 (- x)}{2 \cdot 14} + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.6.2.3

Rewrite the expression.

$3 x^{2} + \frac{- x}{14} + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 2.7

Move the negative in front of the fraction.

$3 x^{2} - \frac{x}{14} + \frac{d}{d x} [8 x] + \frac{d}{d x} [1]$

Step 3

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

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Step 3.1

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

$3 x^{2} - \frac{x}{14} + 8 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2

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

$3 x^{2} - \frac{x}{14} + 8 \cdot 1 + \frac{d}{d x} [1]$

Step 3.3

Multiply $8$ by $1$ .

$3 x^{2} - \frac{x}{14} + 8 + \frac{d}{d x} [1]$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$3 x^{2} - \frac{x}{14} + 8 + 0$

Step 4.2

Add $3 x^{2} - \frac{x}{14} + 8$ and $0$ .

$3 x^{2} - \frac{x}{14} + 8$