Calculus Examples

$y = \frac{u + 1}{u}$ , $u = 2 x^{3}$

Step 1

The chain rule states that the derivative of $y$ with respect to $x$ is equal to the derivative of $y$ with respect to $u$ times the derivative of $u$ with respect to $x$ .

$\frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x}$

Step 2

Find $\frac{d y}{d u}$ .

Tap for more steps...

Step 2.1

Differentiate using the Quotient Rule which states that $\frac{d}{d u} [\frac{f (u)}{g (u)}]$ is $\frac{g (u) \frac{d}{d u} [f (u)] - f (u) \frac{d}{d u} [g (u)]}{{g (u)}^{2}}$ where $f (u) = u + 1$ and $g (u) = u$ .

$\frac{u \frac{d}{d u} [u + 1] - (u + 1) \frac{d}{d u} [u]}{u^{2}}$

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

$\frac{u (\frac{d}{d u} [u] + \frac{d}{d u} [1]) - (u + 1) \frac{d}{d u} [u]}{u^{2}}$

Step 2.2.2

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

$\frac{u (1 + \frac{d}{d u} [1]) - (u + 1) \frac{d}{d u} [u]}{u^{2}}$

Step 2.2.3

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

$\frac{u (1 + 0) - (u + 1) \frac{d}{d u} [u]}{u^{2}}$

Step 2.2.4

Simplify the expression.

Tap for more steps...

Step 2.2.4.1

Add $1$ and $0$ .

$\frac{u \cdot 1 - (u + 1) \frac{d}{d u} [u]}{u^{2}}$

Step 2.2.4.2

Multiply $u$ by $1$ .

$\frac{u - (u + 1) \frac{d}{d u} [u]}{u^{2}}$

Step 2.2.5

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

$\frac{u - (u + 1) \cdot 1}{u^{2}}$

Step 2.2.6

Multiply $- 1$ by $1$ .

$\frac{u - (u + 1)}{u^{2}}$

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Apply the distributive property.

$\frac{u - u - 1 \cdot 1}{u^{2}}$

Step 2.3.2

Simplify the numerator.

Tap for more steps...

Step 2.3.2.1

Subtract $u$ from $u$ .

$\frac{0 - 1 \cdot 1}{u^{2}}$

Step 2.3.2.2

Subtract $1 \cdot 1$ from $0$ .

$\frac{- 1 \cdot 1}{u^{2}}$

Step 2.3.2.3

Multiply $- 1$ by $1$ .

$\frac{- 1}{u^{2}}$

Step 2.3.3

Move the negative in front of the fraction.

$- \frac{1}{u^{2}}$

Step 3

Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

$2 (3 x^{2})$

Step 3.3

Multiply $3$ by $2$ .

$6 x^{2}$

Step 4

Multiply $\frac{d y}{d u}$ by $\frac{d u}{d x}$ .

$\frac{d y}{d x} = (6 x^{2}) \cdot (- \frac{1}{u^{2}})$

Step 5

Simplify the right side $(6 x^{2}) \cdot (- \frac{1}{u^{2}})$ .

Tap for more steps...

Step 5.1

Multiply $(6 x^{2}) (- \frac{1}{u^{2}})$ .

Tap for more steps...

Step 5.1.1

Multiply $- 1$ by $6$ .

$\frac{d y}{d x} = - 6 x^{2} \frac{1}{u^{2}}$

Step 5.1.2

Combine $- 6$ and $\frac{1}{u^{2}}$ .

$\frac{d y}{d x} = x^{2} (\frac{- 6}{u^{2}})$

Step 5.1.3

Combine $x^{2}$ and $\frac{- 6}{u^{2}}$ .

$\frac{d y}{d x} = \frac{x^{2} \cdot - 6}{u^{2}}$

Step 5.2

Move $- 6$ to the left of $x^{2}$ .

$\frac{d y}{d x} = \frac{- 6 \cdot x^{2}}{u^{2}}$

Step 5.3

Move the negative in front of the fraction.

$\frac{d y}{d x} = - \frac{6 x^{2}}{u^{2}}$

Step 6

Substitute in the value of $u$ into the derivative $- \frac{6 x^{2}}{u^{2}}$ .

$\frac{d y}{d x} = - \frac{6 x^{2}}{{(2 x^{3})}^{2}}$

Step 7

Simplify the result.

Tap for more steps...

Step 7.1

Simplify the denominator.

Tap for more steps...

Step 7.1.1

Apply the product rule to $2 x^{3}$ .

$\frac{d y}{d x} = - \frac{6 x^{2}}{2^{2} {(x^{3})}^{2}}$

Step 7.1.2

Raise $2$ to the power of $2$ .

$\frac{d y}{d x} = - \frac{6 x^{2}}{4 {(x^{3})}^{2}}$

Step 7.1.3

Multiply the exponents in ${(x^{3})}^{2}$ .

Tap for more steps...

Step 7.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d y}{d x} = - \frac{6 x^{2}}{4 x^{3 \cdot 2}}$

Step 7.1.3.2

Multiply $3$ by $2$ .

$\frac{d y}{d x} = - \frac{6 x^{2}}{4 x^{6}}$

Step 7.2

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 7.2.1.1

Factor $2$ out of $6 x^{2}$ .

$\frac{d y}{d x} = - \frac{2 (3 x^{2})}{4 x^{6}}$

Step 7.2.1.2

Cancel the common factors.

Tap for more steps...

Step 7.2.1.2.1

Factor $2$ out of $4 x^{6}$ .

$\frac{d y}{d x} = - \frac{2 (3 x^{2})}{2 (2 x^{6})}$

Step 7.2.1.2.2

Cancel the common factor.

$\frac{d y}{d x} = - \frac{2 (3 x^{2})}{2 (2 x^{6})}$

Step 7.2.1.2.3

Rewrite the expression.

$\frac{d y}{d x} = - \frac{3 x^{2}}{2 x^{6}}$

Step 7.2.2

Cancel the common factor of $x^{2}$ and $x^{6}$ .

Tap for more steps...

Step 7.2.2.1

Factor $x^{2}$ out of $3 x^{2}$ .

$\frac{d y}{d x} = - \frac{x^{2} \cdot 3}{2 x^{6}}$

Step 7.2.2.2

Cancel the common factors.

Tap for more steps...

Step 7.2.2.2.1

Factor $x^{2}$ out of $2 x^{6}$ .

$\frac{d y}{d x} = - \frac{x^{2} \cdot 3}{x^{2} (2 x^{4})}$

Step 7.2.2.2.2

Cancel the common factor.

$\frac{d y}{d x} = - \frac{x^{2} \cdot 3}{x^{2} (2 x^{4})}$

Step 7.2.2.2.3

Rewrite the expression.

$\frac{d y}{d x} = - \frac{3}{2 x^{4}}$