Calculus Examples

$y = 3 x^{\frac{2}{3}} - 4 x^{\frac{1}{2}} - 2$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (3 x^{\frac{2}{3}} - 4 x^{\frac{1}{2}} - 2)$

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 $3 x^{\frac{2}{3}} - 4 x^{\frac{1}{2}} - 2$ with respect to $x$ is $\frac{d}{d x} [3 x^{\frac{2}{3}}] + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [3 x^{\frac{2}{3}}] + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2

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

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Step 3.2.1

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

$3 \frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$3 (\frac{2}{3} x^{\frac{2}{3} - 1}) + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}) + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.4

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

$3 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}) + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.5

Combine the numerators over the common denominator.

$3 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}) + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.6

Simplify the numerator.

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Step 3.2.6.1

Multiply $- 1$ by $3$ .

$3 (\frac{2}{3} x^{\frac{2 - 3}{3}}) + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.6.2

Subtract $3$ from $2$ .

$3 (\frac{2}{3} x^{\frac{- 1}{3}}) + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.7

Move the negative in front of the fraction.

$3 (\frac{2}{3} x^{- \frac{1}{3}}) + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.8

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

$3 \frac{2 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.9

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

$\frac{3 (2 x^{- \frac{1}{3}})}{3} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.10

Multiply $2$ by $3$ .

$\frac{6 x^{- \frac{1}{3}}}{3} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.11

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.12

Factor $3$ out of $6$ .

$\frac{3 \cdot 2}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.13

Cancel the common factors.

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Step 3.2.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

$\frac{3 \cdot 2}{3 (x^{\frac{1}{3}})} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.13.2

Cancel the common factor.

$\frac{3 \cdot 2}{3 x^{\frac{1}{3}}} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.2.13.3

Rewrite the expression.

$\frac{2}{x^{\frac{1}{3}}} + \frac{d}{d x} [- 4 x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.3

Evaluate $\frac{d}{d x} [- 4 x^{\frac{1}{2}}]$ .

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Step 3.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{\frac{1}{2}}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{\frac{1}{2}}]$ .

$\frac{2}{x^{\frac{1}{3}}} - 4 \frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- 2]$

Step 3.3.2

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

$\frac{2}{x^{\frac{1}{3}}} - 4 (\frac{1}{2} x^{\frac{1}{2} - 1}) + \frac{d}{d x} [- 2]$

Step 3.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2}{x^{\frac{1}{3}}} - 4 (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}) + \frac{d}{d x} [- 2]$

Step 3.3.4

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

$\frac{2}{x^{\frac{1}{3}}} - 4 (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2]$

Step 3.3.5

Combine the numerators over the common denominator.

$\frac{2}{x^{\frac{1}{3}}} - 4 (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}) + \frac{d}{d x} [- 2]$

Step 3.3.6

Simplify the numerator.

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Step 3.3.6.1

Multiply $- 1$ by $2$ .

$\frac{2}{x^{\frac{1}{3}}} - 4 (\frac{1}{2} x^{\frac{1 - 2}{2}}) + \frac{d}{d x} [- 2]$

Step 3.3.6.2

Subtract $2$ from $1$ .

$\frac{2}{x^{\frac{1}{3}}} - 4 (\frac{1}{2} x^{\frac{- 1}{2}}) + \frac{d}{d x} [- 2]$

Step 3.3.7

Move the negative in front of the fraction.

$\frac{2}{x^{\frac{1}{3}}} - 4 (\frac{1}{2} x^{- \frac{1}{2}}) + \frac{d}{d x} [- 2]$

Step 3.3.8

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

$\frac{2}{x^{\frac{1}{3}}} - 4 \frac{x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2]$

Step 3.3.9

Combine $- 4$ and $\frac{x^{- \frac{1}{2}}}{2}$ .

$\frac{2}{x^{\frac{1}{3}}} + \frac{- 4 x^{- \frac{1}{2}}}{2} + \frac{d}{d x} [- 2]$

Step 3.3.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{x^{\frac{1}{3}}} + \frac{- 4}{2 x^{\frac{1}{2}}} + \frac{d}{d x} [- 2]$

Step 3.3.11

Factor $2$ out of $- 4$ .

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

Step 3.3.12

Cancel the common factors.

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Step 3.3.12.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

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

Step 3.3.12.2

Cancel the common factor.

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

Step 3.3.12.3

Rewrite the expression.

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

Step 3.3.13

Move the negative in front of the fraction.

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

Step 3.4

Differentiate using the Constant Rule.

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Step 3.4.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$\frac{2}{x^{\frac{1}{3}}} - \frac{2}{x^{\frac{1}{2}}} + 0$

Step 3.4.2

Add $\frac{2}{x^{\frac{1}{3}}} - \frac{2}{x^{\frac{1}{2}}}$ and $0$ .

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

Step 4

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

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

Step 5

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

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