Calculus Examples

$y^{3} = x$ ; $(1, 1)$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$3 y^{2}$

Step 3

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

$x'$

Step 4

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

$3 y^{2} = x'$

Step 5

Rewrite the equation as $x' = 3 y^{2}$ .

$x' = 3 y^{2}$

Step 6

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

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

Step 7

Replace $x$ with $1$ and $y$ with $1$ in the expression.

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

Step 8

Simplify the result.

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Step 8.1

One to any power is one.

$3 \cdot 1$

Step 8.2

Multiply $3$ by $1$ .

$3$