Calculus Examples

$f (x) = x^{2} \log_{4} (3 - x)$

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 $g (x) = \log_{4} (3 - x)$ .

$x^{2} \frac{d}{d x} [\log_{4} (3 - x)] + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \log_{4} (x)$ and $g (x) = 3 - x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 - x$ .

$x^{2} (\frac{d}{d u} [\log_{4} (u)] \frac{d}{d x} [3 - x]) + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 2.2

The derivative of $\log_{4} (u)$ with respect to $u$ is $\frac{1}{u \ln (4)}$ .

$x^{2} (\frac{1}{u \ln (4)} \frac{d}{d x} [3 - x]) + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $3 - x$ .

$x^{2} (\frac{1}{(3 - x) \ln (4)} \frac{d}{d x} [3 - x]) + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{(3 - x) \ln (4)}$ and $x^{2}$ .

$\frac{x^{2}}{(3 - x) \ln (4)} \frac{d}{d x} [3 - x] + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.2

By the Sum Rule, the derivative of $3 - x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x]$ .

$\frac{x^{2}}{(3 - x) \ln (4)} (\frac{d}{d x} [3] + \frac{d}{d x} [- x]) + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{x^{2}}{(3 - x) \ln (4)} (0 + \frac{d}{d x} [- x]) + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.4

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

$\frac{x^{2}}{(3 - x) \ln (4)} \frac{d}{d x} [- x] + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.5

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

$\frac{x^{2}}{(3 - x) \ln (4)} (- \frac{d}{d x} [x]) + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.6

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

$\frac{x^{2}}{(3 - x) \ln (4)} (- 1 \cdot 1) + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.7

Combine fractions.

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Step 3.7.1

Multiply $- 1$ by $1$ .

$\frac{x^{2}}{(3 - x) \ln (4)} \cdot - 1 + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.7.2

Combine $\frac{x^{2}}{(3 - x) \ln (4)}$ and $- 1$ .

$\frac{x^{2} \cdot - 1}{(3 - x) \ln (4)} + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.7.3

Simplify the expression.

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Step 3.7.3.1

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

$\frac{- 1 \cdot x^{2}}{(3 - x) \ln (4)} + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.7.3.2

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

$\frac{- x^{2}}{(3 - x) \ln (4)} + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.7.3.3

Move the negative in front of the fraction.

$- \frac{x^{2}}{(3 - x) \ln (4)} + \log_{4} (3 - x) \frac{d}{d x} [x^{2}]$

Step 3.8

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

$- \frac{x^{2}}{(3 - x) \ln (4)} + \log_{4} (3 - x) (2 x)$

Step 3.9

Move $\log_{4} (- x + 3)$ .

$- \frac{x^{2}}{(- x + 3) \ln (4)} + 2 x \log_{4} (- x + 3)$

Step 4

To write $2 x \log_{4} (- x + 3)$ as a fraction with a common denominator, multiply by $\frac{(- x + 3) \ln (4)}{(- x + 3) \ln (4)}$ .

$- \frac{x^{2}}{(- x + 3) \ln (4)} + 2 x \log_{4} (- x + 3) \cdot \frac{(- x + 3) \ln (4)}{(- x + 3) \ln (4)}$

Step 5

Combine $2 x \log_{4} (- x + 3)$ and $\frac{(- x + 3) \ln (4)}{(- x + 3) \ln (4)}$ .

$- \frac{x^{2}}{(- x + 3) \ln (4)} + \frac{2 x \log_{4} (- x + 3) ((- x + 3) \ln (4))}{(- x + 3) \ln (4)}$

Step 6

Combine the numerators over the common denominator.

$\frac{- x^{2} + 2 x \log_{4} (- x + 3) ((- x + 3) \ln (4))}{(- x + 3) \ln (4)}$

Step 7

Simplify.

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Step 7.1

Apply the distributive property.

$\frac{- x^{2} + 2 x \log_{4} (- x + 3) (- x \ln (4) + 3 \ln (4))}{(- x + 3) \ln (4)}$

Step 7.2

Apply the distributive property.

$\frac{- x^{2} + 2 x \log_{4} (- x + 3) (- x \ln (4) + 3 \ln (4))}{- x \ln (4) + 3 \ln (4)}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Simplify each term.

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Step 7.3.1.1

Simplify $2 \log_{4} (- x + 3)$ by moving $2$ inside the logarithm.

$\frac{- x^{2} + x \log_{4} ({(- x + 3)}^{2}) (- x \ln (4) + 3 \ln (4))}{- x \ln (4) + 3 \ln (4)}$

Step 7.3.1.2

Simplify each term.

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Step 7.3.1.2.1

Simplify $3 \ln (4)$ by moving $3$ inside the logarithm.

$\frac{- x^{2} + x \log_{4} ({(- x + 3)}^{2}) (- x \ln (4) + \ln (4^{3}))}{- x \ln (4) + 3 \ln (4)}$

Step 7.3.1.2.2

Raise $4$ to the power of $3$ .

$\frac{- x^{2} + x \log_{4} ({(- x + 3)}^{2}) (- x \ln (4) + \ln (64))}{- x \ln (4) + 3 \ln (4)}$

Step 7.3.1.3

Apply the distributive property.

$\frac{- x^{2} + x \log_{4} ({(- x + 3)}^{2}) (- x \ln (4)) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.3.1.4

Rewrite using the commutative property of multiplication.

$\frac{- x^{2} - \ln (4) x \log_{4} ({(- x + 3)}^{2}) x + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.3.1.5

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

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Step 7.3.1.5.1

Move $x$ .

$\frac{- x^{2} - \ln (4) (x \cdot x) \log_{4} ({(- x + 3)}^{2}) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.3.1.5.2

Multiply $x$ by $x$ .

$\frac{- x^{2} - \ln (4) x^{2} \log_{4} ({(- x + 3)}^{2}) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.3.2

Reorder factors in $- x^{2} - \ln (4) x^{2} \log_{4} ({(- x + 3)}^{2}) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)$ .

$\frac{- x^{2} - x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.4

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) - x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.5

Factor $- 1$ out of $- x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2})$ .

$\frac{- (x^{2}) - (x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2})) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.6

Factor $- 1$ out of $- (x^{2}) - (x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}))$ .

$\frac{- (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2})) + x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{- x \ln (4) + 3 \ln (4)}$

Step 7.7

Factor $- 1$ out of $x \log_{4} ({(- x + 3)}^{2}) \ln (64)$ .

$\frac{- (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2})) - (- x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- x \ln (4) + 3 \ln (4)}$

Step 7.8

Factor $- 1$ out of $- (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2})) - (- x \log_{4} ({(- x + 3)}^{2}) \ln (64))$ .

$\frac{- (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- x \ln (4) + 3 \ln (4)}$

Step 7.9

Rewrite $- (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))$ as $- 1 (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))$ .

$\frac{- 1 (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- x \ln (4) + 3 \ln (4)}$

Step 7.10

Factor $- 1$ out of $- x \ln (4)$ .

$\frac{- 1 (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- (x \ln (4)) + 3 \ln (4)}$

Step 7.11

Factor $- 1$ out of $3 \ln (4)$ .

$\frac{- 1 (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- (x \ln (4)) - (- 3 \ln (4))}$

Step 7.12

Factor $- 1$ out of $- (x \ln (4)) - (- 3 \ln (4))$ .

$\frac{- 1 (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- (x \ln (4) - 3 \ln (4))}$

Step 7.13

Rewrite $- (x \ln (4) - 3 \ln (4))$ as $- 1 (x \ln (4) - 3 \ln (4))$ .

$\frac{- 1 (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- 1 (x \ln (4) - 3 \ln (4))}$

Step 7.14

Cancel the common factor.

$\frac{- 1 (x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64))}{- 1 (x \ln (4) - 3 \ln (4))}$

Step 7.15

Rewrite the expression.

$\frac{x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \log_{4} ({(- x + 3)}^{2}) \ln (64)}{x \ln (4) - 3 \ln (4)}$

Step 7.16

Reorder terms.

$\frac{x^{2} + x^{2} \ln (4) \log_{4} ({(- x + 3)}^{2}) - x \ln (64) \log_{4} ({(- x + 3)}^{2})}{x \ln (4) - 3 \ln (4)}$