Calculus Examples

$x^{2} (2 x - 1)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = 2 x - 1$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $2$ by $1$ .

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

Step 2.5

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

$x^{2} (2 + 0) + (2 x - 1) \frac{d}{d x} [x^{2}]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Add $2$ and $0$ .

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

Step 2.6.2

Move $2$ to the left of $x^{2}$ .

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

Step 2.7

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

$2 x^{2} + (2 x - 1) (2 x)$

Step 2.8

Move $2$ to the left of $2 x - 1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $2$ by $2$ .

$2 x^{2} + 4 x \cdot x + 2 \cdot - 1 x$

Step 3.3.2

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

$2 x^{2} + 4 (x^{1} x) + 2 \cdot - 1 x$

Step 3.3.3

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

$2 x^{2} + 4 (x^{1} x^{1}) + 2 \cdot - 1 x$

Step 3.3.4

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

$2 x^{2} + 4 x^{1 + 1} + 2 \cdot - 1 x$

Step 3.3.5

Add $1$ and $1$ .

$2 x^{2} + 4 x^{2} + 2 \cdot - 1 x$

Step 3.3.6

Multiply $2$ by $- 1$ .

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

Step 3.3.7

Add $2 x^{2}$ and $4 x^{2}$ .

$6 x^{2} - 2 x$