Calculus Examples

$1 + 3 y^{3} = x^{2}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

Differentiate.

Tap for more steps...

Step 2.1.1

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

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

Step 2.1.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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 $g (x) = y$ .

Tap for more steps...

Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

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

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

Step 2.2.3

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

$0 + 3 (3 y^{2} y')$

Step 2.2.4

Multiply $3$ by $3$ .

$0 + 9 y^{2} y'$

Step 2.3

Add $0$ and $9 y^{2} y'$ .

$9 y^{2} y'$

Step 3

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

$2 x$

Step 4

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

$9 y^{2} y' = 2 x$

Step 5

Divide each term in $9 y^{2} y' = 2 x$ by $9 y^{2}$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $9 y^{2} y' = 2 x$ by $9 y^{2}$ .

$\frac{9 y^{2} y'}{9 y^{2}} = \frac{2 x}{9 y^{2}}$

Step 5.2

Simplify the left side.

Tap for more steps...

Step 5.2.1

Cancel the common factor of $9$ .

Tap for more steps...

Step 5.2.1.1

Cancel the common factor.

$\frac{9 y^{2} y'}{9 y^{2}} = \frac{2 x}{9 y^{2}}$

Step 5.2.1.2

Rewrite the expression.

$\frac{y^{2} y'}{y^{2}} = \frac{2 x}{9 y^{2}}$

Step 5.2.2

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

Tap for more steps...

Step 5.2.2.1

Cancel the common factor.

$\frac{y^{2} y'}{y^{2}} = \frac{2 x}{9 y^{2}}$

Step 5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{2 x}{9 y^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{2 x}{9 y^{2}}$