Calculus Examples

${(x^{3} + 3 y)}^{6} = x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ({(x^{3} + 3 y)}^{6}) = \frac{d}{d x} (x)$

Step 2

Differentiate the left side of the equation.

Tap for more steps...

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 $f (x) = x^{6}$ and $g (x) = x^{3} + 3 y$ .

Tap for more steps...

Step 2.1.1

To apply the Chain Rule, set $u$ as $x^{3} + 3 y$ .

$\frac{d}{d u} [u^{6}] \frac{d}{d x} [x^{3} + 3 y]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$6 u^{5} \frac{d}{d x} [x^{3} + 3 y]$

Step 2.1.3

Replace all occurrences of $u$ with $x^{3} + 3 y$ .

$6 {(x^{3} + 3 y)}^{5} \frac{d}{d x} [x^{3} + 3 y]$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

$6 {(x^{3} + 3 y)}^{5} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 y])$

Step 2.2.2

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

$6 {(x^{3} + 3 y)}^{5} (3 x^{2} + \frac{d}{d x} [3 y])$

Step 2.2.3

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

$6 {(x^{3} + 3 y)}^{5} (3 x^{2} + 3 \frac{d}{d x} [y])$

Step 2.3

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

$6 {(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y')$

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.

$6 {(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y') = 1$

Step 5

Solve for $y'$ .

Tap for more steps...

Step 5.1

Divide each term in $6 {(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y') = 1$ by $6 {(x^{3} + 3 y)}^{5}$ and simplify.

Tap for more steps...

Step 5.1.1

Divide each term in $6 {(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y') = 1$ by $6 {(x^{3} + 3 y)}^{5}$ .

$\frac{6 {(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y')}{6 {(x^{3} + 3 y)}^{5}} = \frac{1}{6 {(x^{3} + 3 y)}^{5}}$

Step 5.1.2

Simplify the left side.

Tap for more steps...

Step 5.1.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 5.1.2.1.1

Cancel the common factor.

$\frac{6 {(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y')}{6 {(x^{3} + 3 y)}^{5}} = \frac{1}{6 {(x^{3} + 3 y)}^{5}}$

Step 5.1.2.1.2

Rewrite the expression.

$\frac{{(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y')}{{(x^{3} + 3 y)}^{5}} = \frac{1}{6 {(x^{3} + 3 y)}^{5}}$

Step 5.1.2.2

Cancel the common factor of ${(x^{3} + 3 y)}^{5}$ .

Tap for more steps...

Step 5.1.2.2.1

Cancel the common factor.

$\frac{{(x^{3} + 3 y)}^{5} (3 x^{2} + 3 y')}{{(x^{3} + 3 y)}^{5}} = \frac{1}{6 {(x^{3} + 3 y)}^{5}}$

Step 5.1.2.2.2

Divide $3 x^{2} + 3 y'$ by $1$ .

$3 x^{2} + 3 y' = \frac{1}{6 {(x^{3} + 3 y)}^{5}}$

Step 5.2

Subtract $3 x^{2}$ from both sides of the equation.

$3 y' = \frac{1}{6 {(x^{3} + 3 y)}^{5}} - 3 x^{2}$

Step 5.3

Divide each term in $3 y' = \frac{1}{6 {(x^{3} + 3 y)}^{5}} - 3 x^{2}$ by $3$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $3 y' = \frac{1}{6 {(x^{3} + 3 y)}^{5}} - 3 x^{2}$ by $3$ .

$\frac{3 y'}{3} = \frac{\frac{1}{6 {(x^{3} + 3 y)}^{5}}}{3} + \frac{- 3 x^{2}}{3}$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 y'}{3} = \frac{\frac{1}{6 {(x^{3} + 3 y)}^{5}}}{3} + \frac{- 3 x^{2}}{3}$

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{1}{6 {(x^{3} + 3 y)}^{5}}}{3} + \frac{- 3 x^{2}}{3}$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{\frac{1}{6 {(x^{3} + 3 y)}^{5}} - 3 x^{2}}{3}$

Step 5.3.3.2

Simplify the numerator.

Tap for more steps...

Step 5.3.3.2.1

To write $- 3 x^{2}$ as a fraction with a common denominator, multiply by $\frac{6 {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}}$ .

$y' = \frac{\frac{1}{6 {(x^{3} + 3 y)}^{5}} - 3 x^{2} \cdot \frac{6 {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}}}{3}$

Step 5.3.3.2.2

Combine $- 3 x^{2}$ and $\frac{6 {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}}$ .

$y' = \frac{\frac{1}{6 {(x^{3} + 3 y)}^{5}} + \frac{- 3 x^{2} (6 {(x^{3} + 3 y)}^{5})}{6 {(x^{3} + 3 y)}^{5}}}{3}$

Step 5.3.3.2.3

Combine the numerators over the common denominator.

$y' = \frac{\frac{1 - 3 x^{2} (6 {(x^{3} + 3 y)}^{5})}{6 {(x^{3} + 3 y)}^{5}}}{3}$

Step 5.3.3.2.4

Simplify the numerator.

Tap for more steps...

Step 5.3.3.2.4.1

Rewrite using the commutative property of multiplication.

$y' = \frac{\frac{1 - 3 \cdot 6 x^{2} {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}}}{3}$

Step 5.3.3.2.4.2

Multiply $- 3$ by $6$ .

$y' = \frac{\frac{1 - 18 x^{2} {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}}}{3}$

Step 5.3.3.3

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{1 - 18 x^{2} {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}} \cdot \frac{1}{3}$

Step 5.3.3.4

Multiply $\frac{1 - 18 x^{2} {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 5.3.3.4.1

Multiply $\frac{1 - 18 x^{2} {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5}}$ by $\frac{1}{3}$ .

$y' = \frac{1 - 18 x^{2} {(x^{3} + 3 y)}^{5}}{6 {(x^{3} + 3 y)}^{5} \cdot 3}$

Step 5.3.3.4.2

Multiply $3$ by $6$ .

$y' = \frac{1 - 18 x^{2} {(x^{3} + 3 y)}^{5}}{18 {(x^{3} + 3 y)}^{5}}$

Step 6

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

$\frac{d y}{d x} = \frac{1 - 18 x^{2} {(x^{3} + 3 y)}^{5}}{18 {(x^{3} + 3 y)}^{5}}$