Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

Tap for more steps...

Step 2.1

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

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

Step 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 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Multiply $3$ by $3$ .

$9 (y^{2} \frac{d}{d x} [y])$

Step 2.4

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

$9 y^{2} y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 3.2

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

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

$0 + 3 (2 x)$

Step 3.2.3

Multiply $2$ by $3$ .

$0 + 6 x$

Step 3.3

Add $0$ and $6 x$ .

$6 x$

Step 4

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

$9 y^{2} y' = 6 x$

Step 5

Divide each term in $9 y^{2} y' = 6 x$ by $9 y^{2}$ and simplify.

Tap for more steps...

Step 5.1

Divide each term in $9 y^{2} y' = 6 x$ by $9 y^{2}$ .

$\frac{9 y^{2} y'}{9 y^{2}} = \frac{6 x}{9 y^{2}}$

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.

$\frac{9 y^{2} y'}{9 y^{2}} = \frac{6 x}{9 y^{2}}$

Step 5.2.1.2

Rewrite the expression.

$\frac{y^{2} y'}{y^{2}} = \frac{6 x}{9 y^{2}}$

Step 5.2.2

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

Tap for more steps...

Step 5.2.2.1

Cancel the common factor.

$\frac{y^{2} y'}{y^{2}} = \frac{6 x}{9 y^{2}}$

Step 5.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{6 x}{9 y^{2}}$

Step 5.3

Simplify the right side.

Tap for more steps...

Step 5.3.1

Cancel the common factor of $6$ and $9$ .

Tap for more steps...

Step 5.3.1.1

Factor $3$ out of $6 x$ .

$y' = \frac{3 (2 x)}{9 y^{2}}$

Step 5.3.1.2

Cancel the common factors.

Tap for more steps...

Step 5.3.1.2.1

Factor $3$ out of $9 y^{2}$ .

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

Step 5.3.1.2.2

Cancel the common factor.

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

Step 5.3.1.2.3

Rewrite the expression.

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

Step 6

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

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