Calculus Examples

$y = x^{3}$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

$3 x^{2} x'$

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $3 x^{2} x' = 1$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

Step 5.2.2.2

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

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Step 5.2.2.2.1

Cancel the common factor.

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

Step 5.2.2.2.2

Divide $x'$ by $1$ .

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

Step 6

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

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