Calculus Examples

$y^{3} = x$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.1.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [y]$

Step 2.1.2

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

$3 u^{2} \frac{d}{d x} [y]$

Step 2.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 2.2

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

$3 y^{2} y'$

Step 3

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

$1$

Step 4

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

$3 y^{2} y' = 1$

Step 5

Divide each term in $3 y^{2} y' = 1$ by $3 y^{2}$ and simplify.

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Step 5.1

Divide each term in $3 y^{2} y' = 1$ by $3 y^{2}$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Rewrite the expression.

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

Step 5.2.2

Cancel the common factor of $y^{2}$ .

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Step 5.2.2.1

Cancel the common factor.

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

Step 5.2.2.2

Divide $y'$ by $1$ .

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

Step 6

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

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