Calculus Examples

$y = x - \frac{1}{60} x^{3}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x - \frac{1}{60} x^{3})$

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

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $x - \frac{1}{60} x^{3}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{60} x^{3}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{60} x^{3}]$

Step 3.1.2

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

$1 + \frac{d}{d x} [- \frac{1}{60} x^{3}]$

Step 3.2

Evaluate $\frac{d}{d x} [- \frac{1}{60} x^{3}]$ .

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Step 3.2.1

Since $- \frac{1}{60}$ is constant with respect to $x$ , the derivative of $- \frac{1}{60} x^{3}$ with respect to $x$ is $- \frac{1}{60} \frac{d}{d x} [x^{3}]$ .

$1 - \frac{1}{60} \frac{d}{d x} [x^{3}]$

Step 3.2.2

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

$1 - \frac{1}{60} (3 x^{2})$

Step 3.2.3

Multiply $3$ by $- 1$ .

$1 - 3 (\frac{1}{60}) x^{2}$

Step 3.2.4

Combine $- 3$ and $\frac{1}{60}$ .

$1 + \frac{- 3}{60} x^{2}$

Step 3.2.5

Combine $\frac{- 3}{60}$ and $x^{2}$ .

$1 + \frac{- 3 x^{2}}{60}$

Step 3.2.6

Cancel the common factor of $- 3$ and $60$ .

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Step 3.2.6.1

Factor $3$ out of $- 3 x^{2}$ .

$1 + \frac{3 (- x^{2})}{60}$

Step 3.2.6.2

Cancel the common factors.

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Step 3.2.6.2.1

Factor $3$ out of $60$ .

$1 + \frac{3 (- x^{2})}{3 (20)}$

Step 3.2.6.2.2

Cancel the common factor.

$1 + \frac{3 (- x^{2})}{3 \cdot 20}$

Step 3.2.6.2.3

Rewrite the expression.

$1 + \frac{- x^{2}}{20}$

Step 3.2.7

Move the negative in front of the fraction.

$1 - \frac{x^{2}}{20}$

Step 3.3

Reorder terms.

$- \frac{x^{2}}{20} + 1$

Step 4

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

$y' = - \frac{x^{2}}{20} + 1$

Step 5

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

$\frac{d y}{d x} = - \frac{x^{2}}{20} + 1$