Calculus Examples

$y = x (9 - x^{2})$

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 $f (x) = x$ and $g (x) = 9 - x^{2}$ .

$x \frac{d}{d x} [9 - x^{2}] + (9 - x^{2}) \frac{d}{d x} [x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $9 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]$ .

$x (\frac{d}{d x} [9] + \frac{d}{d x} [- x^{2}]) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 2.2

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$x (0 + \frac{d}{d x} [- x^{2}]) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 2.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

$x \frac{d}{d x} [- x^{2}] + (9 - x^{2}) \frac{d}{d x} [x]$

Step 2.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$x (- \frac{d}{d x} [x^{2}]) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 2.5

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

$x (- (2 x)) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 2.6

Multiply $2$ by $- 1$ .

$x (- 2 x) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 3

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

$- 2 (x^{1} x) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 4

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

$- 2 (x^{1} x^{1}) + (9 - x^{2}) \frac{d}{d x} [x]$

Step 5

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

$- 2 x^{1 + 1} + (9 - x^{2}) \frac{d}{d x} [x]$

Step 6

Add $1$ and $1$ .

$- 2 x^{2} + (9 - x^{2}) \frac{d}{d x} [x]$

Step 7

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

$- 2 x^{2} + (9 - x^{2}) \cdot 1$

Step 8

Simplify by adding terms.

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Step 8.1

Multiply $9 - x^{2}$ by $1$ .

$- 2 x^{2} + 9 - x^{2}$

Step 8.2

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

$- 3 x^{2} + 9$