Calculus Examples

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Since $- 2$ is constant with respect to $u$ , the derivative of $- 2$ with respect to $u$ is $0$ .

$2 u + 1 + 0$

Step 2.5

Add $2 u + 1$ and $0$ .

$2 u + 1$

Step 3

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

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Step 3.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 3.2

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

$- x^{- 2}$

Step 3.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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Step 5.2.1

Multiply $2$ by $- 1$ .

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

Multiply $- 1$ by $1$ .

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

Step 5.4

Move the negative in front of the fraction.

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

Step 6

Substitute in the value of $u$ into the derivative $- \frac{2 u}{x^{2}} - \frac{1}{x^{2}}$ .

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

Step 7

Simplify each term.

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Step 7.1

Combine $2$ and $\frac{1}{x}$ .

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

Step 7.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.3

Combine.

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

Step 7.4

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

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Step 7.4.1

Multiply $x$ by $x^{2}$ .

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Step 7.4.1.1

Raise $x$ to the power of $1$ .

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

Step 7.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.4.2

Add $1$ and $2$ .

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

Step 7.5

Multiply $2$ by $1$ .

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