Calculus Examples

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

Step 1

Differentiate.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Evaluate $\frac{d}{d x} [- 2 x^{2}]$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $2$ by $- 2$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Add $3 x^{2} - 4 x + 1$ and $0$ .

$3 x^{2} - 4 x + 1$