Calculus Examples

$f (x) = - 4096 + 64 (x^{\frac{2}{3}}) y^{\frac{1}{3}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $- 4096 + 64 (x^{\frac{2}{3}}) y^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [- 4096] + \frac{d}{d x} [64 (x^{\frac{2}{3}}) y^{\frac{1}{3}}]$ .

$\frac{d}{d x} [- 4096] + \frac{d}{d x} [64 (x^{\frac{2}{3}}) y^{\frac{1}{3}}]$

Step 1.2

Since $- 4096$ is constant with respect to $x$ , the derivative of $- 4096$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [64 (x^{\frac{2}{3}}) y^{\frac{1}{3}}]$

Step 2

Evaluate $\frac{d}{d x} [64 (x^{\frac{2}{3}}) y^{\frac{1}{3}}]$ .

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Step 2.1

Since $64 y^{\frac{1}{3}}$ is constant with respect to $x$ , the derivative of $64 x^{\frac{2}{3}} y^{\frac{1}{3}}$ with respect to $x$ is $64 y^{\frac{1}{3}} \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$0 + 64 y^{\frac{1}{3}} \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 2.2

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

$0 + 64 y^{\frac{1}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 2.3

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

$0 + 64 y^{\frac{1}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 2.4

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

$0 + 64 y^{\frac{1}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 2.5

Combine the numerators over the common denominator.

$0 + 64 y^{\frac{1}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

$0 + 64 y^{\frac{1}{3}} (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 2.6.2

Subtract $3$ from $2$ .

$0 + 64 y^{\frac{1}{3}} (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 2.7

Move the negative in front of the fraction.

$0 + 64 y^{\frac{1}{3}} (\frac{2}{3} x^{- \frac{1}{3}})$

Step 2.8

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

$0 + 64 y^{\frac{1}{3}} \frac{2 x^{- \frac{1}{3}}}{3}$

Step 2.9

Combine $64$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$0 + y^{\frac{1}{3}} \frac{64 (2 x^{- \frac{1}{3}})}{3}$

Step 2.10

Multiply $2$ by $64$ .

$0 + y^{\frac{1}{3}} \frac{128 x^{- \frac{1}{3}}}{3}$

Step 2.11

Combine $y^{\frac{1}{3}}$ and $\frac{128 x^{- \frac{1}{3}}}{3}$ .

$0 + \frac{y^{\frac{1}{3}} (128 x^{- \frac{1}{3}})}{3}$

Step 2.12

Move $128$ to the left of $y^{\frac{1}{3}}$ .

$0 + \frac{128 \cdot y^{\frac{1}{3}} x^{- \frac{1}{3}}}{3}$

Step 2.13

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

$0 + \frac{128 y^{\frac{1}{3}}}{3 x^{\frac{1}{3}}}$

Step 3

Add $0$ and $\frac{128 y^{\frac{1}{3}}}{3 x^{\frac{1}{3}}}$ .

$\frac{128 y^{\frac{1}{3}}}{3 x^{\frac{1}{3}}}$