Calculus Examples

$\frac{f (x) (a x + b x^{2})}{a m + b m^{2}}$

Step 1

Factor $x$ out of $a x + b x^{2}$ .

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Step 1.1

Factor $x$ out of $a x$ .

$\frac{d}{d x} [\frac{f (x) (x a + b x^{2})}{a m + b m^{2}}]$

Step 1.2

Factor $x$ out of $b x^{2}$ .

$\frac{d}{d x} [\frac{f (x) (x a + x (b x))}{a m + b m^{2}}]$

Step 1.3

Factor $x$ out of $x a + x (b x)$ .

$\frac{d}{d x} [\frac{f (x) (x (a + b x))}{a m + b m^{2}}]$

$\frac{d}{d x} [\frac{f (x) x (a + b x)}{a m + b m^{2}}]$

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Factor $m$ out of $a m + b m^{2}$ .

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Step 2.1.1

Factor $m$ out of $a m$ .

$\frac{d}{d x} [\frac{f (x) x (a + b x)}{m a + b m^{2}}]$

Step 2.1.2

Factor $m$ out of $b m^{2}$ .

$\frac{d}{d x} [\frac{f (x) x (a + b x)}{m a + m (b m)}]$

Step 2.1.3

Factor $m$ out of $m a + m (b m)$ .

$\frac{d}{d x} [\frac{f (x) x (a + b x)}{m (a + b m)}]$

Step 2.2

Reorder factors in $\frac{f (x) x (a + b x)}{m (a + b m)}$ .

$\frac{d}{d x} [\frac{x (a + b x) f (x)}{m (a + b m)}]$

Step 2.3

Since $\frac{f (x)}{m (a + b m)}$ is constant with respect to $x$ , the derivative of $\frac{x (a + b x) f (x)}{m (a + b m)}$ with respect to $x$ is $\frac{f (x)}{m (a + b m)} \frac{d}{d x} [x (a + b x)]$ .

$\frac{f (x)}{m (a + b m)} \frac{d}{d x} [x (a + b x)]$

Step 3

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 $g (x) = a + b x$ .

$\frac{f (x)}{m (a + b m)} (x \frac{d}{d x} [a + b x] + (a + b x) \frac{d}{d x} [x])$

Step 4

Differentiate.

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Step 4.1

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

$\frac{f (x)}{m (a + b m)} (x (\frac{d}{d x} [a] + \frac{d}{d x} [b x]) + (a + b x) \frac{d}{d x} [x])$

Step 4.2

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

$\frac{f (x)}{m (a + b m)} (x (0 + \frac{d}{d x} [b x]) + (a + b x) \frac{d}{d x} [x])$

Step 4.3

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

$\frac{f (x)}{m (a + b m)} (x \frac{d}{d x} [b x] + (a + b x) \frac{d}{d x} [x])$

Step 4.4

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

$\frac{f (x)}{m (a + b m)} (x (b \frac{d}{d x} [x]) + (a + b x) \frac{d}{d x} [x])$

Step 4.5

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

$\frac{f (x)}{m (a + b m)} (x (b \cdot 1) + (a + b x) \frac{d}{d x} [x])$

Step 4.6

Multiply $b$ by $1$ .

$\frac{f (x)}{m (a + b m)} (x b + (a + b x) \frac{d}{d x} [x])$

Step 4.7

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

$\frac{f (x)}{m (a + b m)} (x b + (a + b x) \cdot 1)$

Step 4.8

Multiply $a + b x$ by $1$ .

$\frac{f (x)}{m (a + b m)} (x b + a + b x)$

Step 5

Add $x b$ and $b x$ .

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Step 5.1

Reorder $x$ and $b$ .

$\frac{f (x)}{m (a + b m)} (a + b x + b x)$

Step 5.2

Add $b x$ and $b x$ .

$\frac{f (x)}{m (a + b m)} (a + 2 b x)$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{f (x)}{m a + m (b m)} (a + 2 b x)$

Step 6.2

Apply the distributive property.

$\frac{f (x)}{m a + m (b m)} a + \frac{f (x)}{m a + m (b m)} (2 b x)$

Step 6.3

Combine terms.

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Step 6.3.1

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

$\frac{f (x)}{m a + b (m^{1} m)} a + \frac{f (x)}{m a + m (b m)} (2 b x)$

Step 6.3.2

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

$\frac{f (x)}{m a + b (m^{1} m^{1})} a + \frac{f (x)}{m a + m (b m)} (2 b x)$

Step 6.3.3

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

$\frac{f (x)}{m a + b m^{1 + 1}} a + \frac{f (x)}{m a + m (b m)} (2 b x)$

Step 6.3.4

Add $1$ and $1$ .

$\frac{f (x)}{m a + b m^{2}} a + \frac{f (x)}{m a + m (b m)} (2 b x)$

Step 6.3.5

Combine $\frac{f (x)}{m a + b m^{2}}$ and $a$ .

$\frac{f (x) a}{m a + b m^{2}} + \frac{f (x)}{m a + m (b m)} (2 b x)$

Step 6.3.6

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

$\frac{f (x) a}{m a + b m^{2}} + \frac{f (x)}{m a + b (m^{1} m)} (2 b x)$

Step 6.3.7

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

$\frac{f (x) a}{m a + b m^{2}} + \frac{f (x)}{m a + b (m^{1} m^{1})} (2 b x)$

Step 6.3.8

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

$\frac{f (x) a}{m a + b m^{2}} + \frac{f (x)}{m a + b m^{1 + 1}} (2 b x)$

Step 6.3.9

Add $1$ and $1$ .

$\frac{f (x) a}{m a + b m^{2}} + \frac{f (x)}{m a + b m^{2}} (2 b x)$

Step 6.3.10

Combine $2$ and $\frac{f (x)}{m a + b m^{2}}$ .

$\frac{f (x) a}{m a + b m^{2}} + \frac{2 f (x)}{m a + b m^{2}} (b x)$

Step 6.3.11

Combine $b$ and $\frac{2 f (x)}{m a + b m^{2}}$ .

$\frac{f (x) a}{m a + b m^{2}} + \frac{b (2 f (x))}{m a + b m^{2}} x$

Step 6.3.12

Combine $\frac{b (2 f (x))}{m a + b m^{2}}$ and $x$ .

$\frac{f (x) a}{m a + b m^{2}} + \frac{b (2 f (x)) x}{m a + b m^{2}}$

Step 6.3.13

Move $2$ to the left of $b$ .

$\frac{f (x) a}{m a + b m^{2}} + \frac{2 \cdot b f (x) x}{m a + b m^{2}}$

Step 6.3.14

Combine the numerators over the common denominator.

$\frac{f (x) a + 2 b f (x) x}{m a + b m^{2}}$

Step 6.4

Reorder terms.

$\frac{a f (x) + 2 b x f (x)}{a m + m^{2} b}$