Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{2} - 3 x + 1$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} - 3 x + 1$ .

$\frac{d}{d u} [u^{5}] \frac{d}{d x} [x^{2} - 3 x + 1]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$5 u^{4} \frac{d}{d x} [x^{2} - 3 x + 1]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} - 3 x + 1$ .

$5 {(x^{2} - 3 x + 1)}^{4} \frac{d}{d x} [x^{2} - 3 x + 1]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - 3 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1]$ .

$5 {(x^{2} - 3 x + 1)}^{4} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1])$

Step 2.2

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

$5 {(x^{2} - 3 x + 1)}^{4} (2 x + \frac{d}{d x} [- 3 x] + \frac{d}{d x} [1])$

Step 3

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

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Step 3.1

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

$5 {(x^{2} - 3 x + 1)}^{4} (2 x - 3 \frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 3.2

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

$5 {(x^{2} - 3 x + 1)}^{4} (2 x - 3 \cdot 1 + \frac{d}{d x} [1])$

Step 3.3

Multiply $- 3$ by $1$ .

$5 {(x^{2} - 3 x + 1)}^{4} (2 x - 3 + \frac{d}{d x} [1])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

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

$5 {(x^{2} - 3 x + 1)}^{4} (2 x - 3 + 0)$

Step 4.2

Add $2 x - 3$ and $0$ .

$5 {(x^{2} - 3 x + 1)}^{4} (2 x - 3)$