Calculus Examples

y = u^{2} + u - 2

$y = u^{2} + u - 2$ ,

u = \frac{1}{x}

$u = \frac{1}{x}$

Step 1

The chain rule states that the derivative of

y

$y$ with respect to

x

$x$ is equal to the derivative of

y

$y$ with respect to

u

$u$ times the derivative of

u

$u$ with respect to

x

$x$ .

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

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

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of

u^{2} + u - 2

$u^{2} + u - 2$ with respect to

u

$u$ is

\frac{d}{d u} [u^{2}] + \frac{d}{d u} [u] + \frac{d}{d u} [- 2]

$\frac{d}{d u} [u^{2}] + \frac{d}{d u} [u] + \frac{d}{d u} [- 2]$ .

\frac{d}{d u} [u^{2}] + \frac{d}{d u} [u] + \frac{d}{d u} [- 2]

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

Step 2.2

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = 2

$n = 2$ .

2 u + \frac{d}{d u} [u] + \frac{d}{d u} [- 2]

$2 u + \frac{d}{d u} [u] + \frac{d}{d u} [- 2]$

Step 2.3

Differentiate using the Power Rule which states that

\frac{d}{d u} [u^{n}]

$\frac{d}{d u} [u^{n}]$ is

n u^{n - 1}

$n u^{n - 1}$ where

n = 1

$n = 1$ .

2 u + 1 + \frac{d}{d u} [- 2]

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

Step 2.4

Since

- 2

$- 2$ is constant with respect to

u

$u$ , the derivative of

- 2

$- 2$ with respect to

u

$u$ is

0

$0$ .

2 u + 1 + 0

$2 u + 1 + 0$

Step 2.5

Add

2 u + 1

$2 u + 1$ and

0

$0$ .

2 u + 1

$2 u + 1$

2 u + 1

$2 u + 1$

Step 3

Find

\frac{d u}{d x}

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

Tap for more steps...

Step 3.1

Rewrite

\frac{1}{x}

$\frac{1}{x}$ as

x^{- 1}

$x^{- 1}$ .

\frac{d}{d x} [x^{- 1}]

$\frac{d}{d x} [x^{- 1}]$

Step 3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = - 1

$n = - 1$ .

- x^{- 2}

$- x^{- 2}$

Step 3.3

Rewrite the expression using the negative exponent rule

b^{- n} = \frac{1}{b^{n}}

$b^{- n} = \frac{1}{b^{n}}$ .

- \frac{1}{x^{2}}

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

- \frac{1}{x^{2}}

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

Step 4

Multiply

\frac{d y}{d u}

$\frac{d y}{d u}$ by

\frac{d u}{d x}

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

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

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

Step 5

Simplify the right side

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

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

Tap for more steps...

Step 5.1

Apply the distributive property.

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

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

Step 5.2

Multiply

- \frac{1}{x^{2}} (2 u)

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

Tap for more steps...

Step 5.2.1

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 5.2.2

Combine

- 2

$- 2$ and

\frac{1}{x^{2}}

$\frac{1}{x^{2}}$ .

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

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

Step 5.2.3

Combine

\frac{- 2}{x^{2}}

$\frac{- 2}{x^{2}}$ and

u

$u$ .

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

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

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

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

Step 5.3

Multiply

- 1

$- 1$ by

1

$1$ .

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

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

Step 5.4

Move the negative in front of the fraction.

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

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

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

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

Step 6

Substitute in the value of

u

$u$ into the derivative

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

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

\frac{d y}{d x} = - \frac{2 (\frac{1}{x})}{x^{2}} - \frac{1}{x^{2}}

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

Step 7

Simplify each term.

Tap for more steps...

Step 7.1

Combine

2

$2$ and

\frac{1}{x}

$\frac{1}{x}$ .

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

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

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

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

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

Step 7.3

Combine.

\frac{d y}{d x} = - \frac{2 \cdot 1}{x \cdot x^{2}} - \frac{1}{x^{2}}

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

Step 7.4

Multiply

x

$x$ by

x^{2}

$x^{2}$ by adding the exponents.

Tap for more steps...

Step 7.4.1

Multiply

x

$x$ by

x^{2}

$x^{2}$ .

Tap for more steps...

Step 7.4.1.1

Raise

x

$x$ to the power of

1

$1$ .

\frac{d y}{d x} = - \frac{2 \cdot 1}{x \cdot x^{2}} - \frac{1}{x^{2}}

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

Step 7.4.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{d y}{d x} = - \frac{2 \cdot 1}{x^{1 + 2}} - \frac{1}{x^{2}}

$\frac{d y}{d x} = - \frac{2 \cdot 1}{x^{1 + 2}} - \frac{1}{x^{2}}$

\frac{d y}{d x} = - \frac{2 \cdot 1}{x^{1 + 2}} - \frac{1}{x^{2}}

$\frac{d y}{d x} = - \frac{2 \cdot 1}{x^{1 + 2}} - \frac{1}{x^{2}}$

Step 7.4.2

Add

1

$1$ and

2

$2$ .

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

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

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

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

Step 7.5

Multiply

2

$2$ by

1

$1$ .

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

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

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

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