Calculus Examples

$y = x^{2} (6 - x)$

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 $g (x) = 6 - x$ .

$x^{2} \frac{d}{d x} [6 - x] + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $6 - x$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [- x]$ .

$x^{2} (\frac{d}{d x} [6] + \frac{d}{d x} [- x]) + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.2

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$x^{2} (0 + \frac{d}{d x} [- x]) + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$x^{2} \frac{d}{d x} [- x] + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$x^{2} (- \frac{d}{d x} [x]) + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.5

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

$x^{2} (- 1 \cdot 1) + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.6

Simplify the expression.

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Step 2.6.1

Multiply $- 1$ by $1$ .

$x^{2} \cdot - 1 + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.6.2

Move $- 1$ to the left of $x^{2}$ .

$- 1 \cdot x^{2} + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.6.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$- x^{2} + (6 - x) \frac{d}{d x} [x^{2}]$

Step 2.7

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

$- x^{2} + (6 - x) (2 x)$

Step 2.8

Move $2$ to the left of $6 - x$ .

$- x^{2} + 2 \cdot (6 - x) x$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$- x^{2} + (2 \cdot 6 + 2 (- x)) x$

Step 3.2

Apply the distributive property.

$- x^{2} + 2 \cdot 6 x + 2 (- x) x$

Step 3.3

Combine terms.

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Step 3.3.1

Multiply $2$ by $6$ .

$- x^{2} + 12 x + 2 (- x) x$

Step 3.3.2

Multiply $- 1$ by $2$ .

$- x^{2} + 12 x - 2 x \cdot x$

Step 3.3.3

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

$- x^{2} + 12 x - 2 (x^{1} x)$

Step 3.3.4

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

$- x^{2} + 12 x - 2 (x^{1} x^{1})$

Step 3.3.5

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

$- x^{2} + 12 x - 2 x^{1 + 1}$

Step 3.3.6

Add $1$ and $1$ .

$- x^{2} + 12 x - 2 x^{2}$

Step 3.3.7

Subtract $2 x^{2}$ from $- x^{2}$ .

$- 3 x^{2} + 12 x$