Calculus Examples

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

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 $u^{2} + 1$ with respect to $u$ is $\frac{d}{d u} [u^{2}] + \frac{d}{d u} [1]$ .

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

Step 2.2

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

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

Step 2.3

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

$2 u + 0$

Step 2.4

Add $2 u$ and $0$ .

$2 u$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 2]$

Step 3.2.3

Multiply $3$ by $1$ .

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

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

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

$3 + 0$

Step 3.3.2

Add $3$ and $0$ .

$3$

Step 4

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

$\frac{d y}{d x} = (3) \cdot (2 u)$

Step 5

Multiply $2$ by $3$ .

$\frac{d y}{d x} = 6 u$

Step 6

Substitute in the value of $u$ into the derivative $6 u$ .

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

Step 7

Simplify the result.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply.

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Step 7.2.1

Multiply $3$ by $6$ .

$\frac{d y}{d x} = 18 x + 6 \cdot - 2$

Step 7.2.2

Multiply $6$ by $- 2$ .

$\frac{d y}{d x} = 18 x - 12$