Calculus Examples

$h (x) = x^{2} \log_{3} (x - π)$

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 $g (x) = \log_{3} (x - π)$ .

$x^{2} \frac{d}{d x} [\log_{3} (x - π)] + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{3} (x)$ and $g (x) = x - π$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x - π$ .

$x^{2} (\frac{d}{d u} [\log_{3} (u)] \frac{d}{d x} [x - π]) + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 2.2

The derivative of $\log_{3} (u)$ with respect to $u$ is $\frac{1}{u \ln (3)}$ .

$x^{2} (\frac{1}{u \ln (3)} \frac{d}{d x} [x - π]) + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $x - π$ .

$x^{2} (\frac{1}{(x - π) \ln (3)} \frac{d}{d x} [x - π]) + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{(x - π) \ln (3)}$ and $x^{2}$ .

$\frac{x^{2}}{(x - π) \ln (3)} \frac{d}{d x} [x - π] + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 3.2

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

$\frac{x^{2}}{(x - π) \ln (3)} (\frac{d}{d x} [x] + \frac{d}{d x} [- π]) + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 3.3

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

$\frac{x^{2}}{(x - π) \ln (3)} (1 + \frac{d}{d x} [- π]) + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 3.4

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

$\frac{x^{2}}{(x - π) \ln (3)} (1 + 0) + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

$\frac{x^{2}}{(x - π) \ln (3)} \cdot 1 + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 3.5.2

Multiply $\frac{x^{2}}{(x - π) \ln (3)}$ by $1$ .

$\frac{x^{2}}{(x - π) \ln (3)} + \log_{3} (x - π) \frac{d}{d x} [x^{2}]$

Step 3.6

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

$\frac{x^{2}}{(x - π) \ln (3)} + \log_{3} (x - π) (2 x)$

Step 4

Combine $\frac{x^{2}}{(x - π) \ln (3)}$ and $\log_{3} (x - π) (2 x)$ using a common denominator.

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Step 4.1

Move $\log_{3} (x - π)$ .

$\frac{x^{2}}{(x - π) \ln (3)} + 2 x \log_{3} (x - π)$

Step 4.2

To write $2 x \log_{3} (x - π)$ as a fraction with a common denominator, multiply by $\frac{(x - π) \ln (3)}{(x - π) \ln (3)}$ .

$\frac{x^{2}}{(x - π) \ln (3)} + 2 x \log_{3} (x - π) \cdot \frac{(x - π) \ln (3)}{(x - π) \ln (3)}$

Step 4.3

Combine $2 x \log_{3} (x - π)$ and $\frac{(x - π) \ln (3)}{(x - π) \ln (3)}$ .

$\frac{x^{2}}{(x - π) \ln (3)} + \frac{2 x \log_{3} (x - π) ((x - π) \ln (3))}{(x - π) \ln (3)}$

Step 4.4

Combine the numerators over the common denominator.

$\frac{x^{2} + 2 x \log_{3} (x - π) ((x - π) \ln (3))}{(x - π) \ln (3)}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{x^{2} + 2 x \log_{3} (x - π) (x \ln (3) - π \ln (3))}{(x - π) \ln (3)}$

Step 5.2

Apply the distributive property.

$\frac{x^{2} + 2 x \log_{3} (x - π) (x \ln (3) - π \ln (3))}{x \ln (3) - π \ln (3)}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Simplify $2 \log_{3} (x - π)$ by moving $2$ inside the logarithm.

$\frac{x^{2} + x \log_{3} ({(x - π)}^{2}) (x \ln (3) - π \ln (3))}{x \ln (3) - π \ln (3)}$

Step 5.3.1.2

Apply the distributive property.

$\frac{x^{2} + x \log_{3} ({(x - π)}^{2}) (x \ln (3)) + x \log_{3} ({(x - π)}^{2}) (- π \ln (3))}{x \ln (3) - π \ln (3)}$

Step 5.3.1.3

Rewrite using the commutative property of multiplication.

$\frac{x^{2} + \ln (3) x \log_{3} ({(x - π)}^{2}) x + x \log_{3} ({(x - π)}^{2}) (- π \ln (3))}{x \ln (3) - π \ln (3)}$

Step 5.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.1.4.1

Move $x$ .

$\frac{x^{2} + \ln (3) (x \cdot x) \log_{3} ({(x - π)}^{2}) + x \log_{3} ({(x - π)}^{2}) (- π \ln (3))}{x \ln (3) - π \ln (3)}$

Step 5.3.1.4.2

Multiply $x$ by $x$ .

$\frac{x^{2} + \ln (3) x^{2} \log_{3} ({(x - π)}^{2}) + x \log_{3} ({(x - π)}^{2}) (- π \ln (3))}{x \ln (3) - π \ln (3)}$

Step 5.3.2

Reorder factors in $x^{2} + \ln (3) x^{2} \log_{3} ({(x - π)}^{2}) + x \log_{3} ({(x - π)}^{2}) (- π \ln (3))$ .

$\frac{x^{2} + x^{2} \ln (3) \log_{3} ({(x - π)}^{2}) - x π \log_{3} ({(x - π)}^{2}) \ln (3)}{x \ln (3) - π \ln (3)}$

Step 5.4

Reorder terms.

$\frac{x^{2} + x^{2} \ln (3) \log_{3} ({(x - π)}^{2}) - π x \ln (3) \log_{3} ({(x - π)}^{2})}{x \ln (3) - π \ln (3)}$