Calculus Examples

$y = x^{n} {(a + b x)}^{m}$

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 $f (x) = x^{n}$ and $g (x) = {(a + b x)}^{m}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{m}$ and $g (x) = a + b x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $a + b x$ .

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

Step 2.2

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

$x^{n} (m u^{m - 1} \frac{d}{d x} [a + b x]) + {(a + b x)}^{m} \frac{d}{d x} [x^{n}]$

Step 2.3

Replace all occurrences of $u$ with $a + b x$ .

$x^{n} (m {(a + b x)}^{m - 1} \frac{d}{d x} [a + b x]) + {(a + b x)}^{m} \frac{d}{d x} [x^{n}]$

Step 3

Differentiate.

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Step 3.1

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

$x^{n} (m {(a + b x)}^{m - 1} (\frac{d}{d x} [a] + \frac{d}{d x} [b x])) + {(a + b x)}^{m} \frac{d}{d x} [x^{n}]$

Step 3.2

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

$x^{n} (m {(a + b x)}^{m - 1} (0 + \frac{d}{d x} [b x])) + {(a + b x)}^{m} \frac{d}{d x} [x^{n}]$

Step 3.3

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

$x^{n} (m {(a + b x)}^{m - 1} \frac{d}{d x} [b x]) + {(a + b x)}^{m} \frac{d}{d x} [x^{n}]$

Step 3.4

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

$x^{n} (m {(a + b x)}^{m - 1} (b \frac{d}{d x} [x])) + {(a + b x)}^{m} \frac{d}{d x} [x^{n}]$

Step 3.5

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

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

Step 3.6

Multiply $b$ by $1$ .

$x^{n} (m {(a + b x)}^{m - 1} b) + {(a + b x)}^{m} \frac{d}{d x} [x^{n}]$

Step 3.7

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

$x^{n} (m {(a + b x)}^{m - 1} b) + {(a + b x)}^{m} (n x^{n - 1})$

Step 4

Simplify.

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Step 4.1

Reorder terms.

$x^{n} b m {(a + b x)}^{m - 1} + x^{n - 1} n {(a + b x)}^{m}$

Step 4.2

Reorder factors in $x^{n} b m {(a + b x)}^{m - 1} + x^{n - 1} n {(a + b x)}^{m}$ .

$b m x^{n} {(a + b x)}^{m - 1} + n x^{n - 1} {(a + b x)}^{m}$