Calculus Examples

$y = 302.2 (\frac{103^{x}}{100^{x}})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $302.2$ and $\frac{103^{x}}{100^{x}}$ .

$\frac{d}{d x} [\frac{302.2 \cdot 103^{x}}{100^{x}}]$

Step 1.2

Since $302.2$ is constant with respect to $x$ , the derivative of $\frac{302.2 \cdot 103^{x}}{100^{x}}$ with respect to $x$ is $302.2 \frac{d}{d x} [\frac{103^{x}}{100^{x}}]$ .

$302.2 \frac{d}{d x} [\frac{103^{x}}{100^{x}}]$

Step 2

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 103^{x}$ and $g (x) = 100^{x}$ .

$302.2 \frac{100^{x} \frac{d}{d x} [103^{x}] - 103^{x} \frac{d}{d x} [100^{x}]}{{(100^{x})}^{2}}$

Step 3

Multiply the exponents in ${(100^{x})}^{2}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$302.2 \frac{100^{x} \frac{d}{d x} [103^{x}] - 103^{x} \frac{d}{d x} [100^{x}]}{100^{x \cdot 2}}$

Step 3.2

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

$302.2 \frac{100^{x} \frac{d}{d x} [103^{x}] - 103^{x} \frac{d}{d x} [100^{x}]}{100^{2 x}}$

Step 4

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $103$ .

$302.2 \frac{100^{x} (103^{x} \ln (103)) - 103^{x} \frac{d}{d x} [100^{x}]}{100^{2 x}}$

Step 5

Simplify with factoring out.

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Step 5.1

Rewrite $103^{x} \cdot 100^{x}$ as ${(103 \cdot 100)}^{x}$ .

$302.2 \frac{{(103 \cdot 100)}^{x} \ln (103) - 103^{x} \frac{d}{d x} [100^{x}]}{100^{2 x}}$

Step 5.2

Multiply $103$ by $100$ .

$302.2 \frac{10300^{x} \ln (103) - 103^{x} \frac{d}{d x} [100^{x}]}{100^{2 x}}$

Step 6

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $100$ .

$302.2 \frac{10300^{x} \ln (103) - 103^{x} (100^{x} \ln (100))}{100^{2 x}}$

Step 7

Simplify terms.

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Step 7.1

Rewrite $100^{x} \cdot 103^{x}$ as ${(100 \cdot 103)}^{x}$ .

$302.2 \frac{10300^{x} \ln (103) - {(100 \cdot 103)}^{x} \ln (100)}{100^{2 x}}$

Step 7.2

Multiply $100$ by $103$ .

$302.2 \frac{10300^{x} \ln (103) - 10300^{x} \ln (100)}{100^{2 x}}$

Step 7.3

Combine $302.2$ and $\frac{10300^{x} \ln (103) - 10300^{x} \ln (100)}{100^{2 x}}$ .

$\frac{302.2 (10300^{x} \ln (103) - 10300^{x} \ln (100))}{100^{2 x}}$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{302.2 (10300^{x} \ln (103)) + 302.2 (- 10300^{x} \ln (100))}{100^{2 x}}$

Step 8.2

Simplify the numerator.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Multiply $302.2 (10300^{x} \ln (103))$ .

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Step 8.2.1.1.1

Reorder $\ln (103)$ and $302.2$ .

$\frac{302.2 \cdot \ln (103) \cdot 10300^{x} + 302.2 (- 10300^{x} \ln (100))}{100^{2 x}}$

Step 8.2.1.1.2

Simplify $302.2 \cdot \ln (103)$ by moving $302.2$ inside the logarithm.

$\frac{\ln (103^{302.2}) \cdot 10300^{x} + 302.2 (- 10300^{x} \ln (100))}{100^{2 x}}$

Step 8.2.1.2

Multiply $302.2 (- 10300^{x} \ln (100))$ .

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Step 8.2.1.2.1

Multiply $- 1$ by $302.2$ .

$\frac{\ln (103^{302.2}) \cdot 10300^{x} - 302.2 (10300^{x} \ln (100))}{100^{2 x}}$

Step 8.2.1.2.2

Reorder $\ln (100)$ and $- 302.2$ .

$\frac{\ln (103^{302.2}) \cdot 10300^{x} - 302.2 \cdot \ln (100) \cdot 10300^{x}}{100^{2 x}}$

Step 8.2.1.2.3

Simplify $- 302.2 \cdot \ln (100)$ by moving $302.2$ inside the logarithm.

$\frac{\ln (103^{302.2}) \cdot 10300^{x} - \ln (100^{302.2}) \cdot 10300^{x}}{100^{2 x}}$

Step 8.2.2

Reorder factors in $\ln (103^{302.2}) \cdot 10300^{x} - \ln (100^{302.2}) \cdot 10300^{x}$ .

$\frac{10300^{x} \ln (103^{302.2}) - 10300^{x} \ln (100^{302.2})}{100^{2 x}}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Factor $10300^{x}$ out of $10300^{x} \ln (103^{302.2}) - 10300^{x} \ln (100^{302.2})$ .

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Step 8.3.1.1

Factor $10300^{x}$ out of $10300^{x} \ln (103^{302.2})$ .

$\frac{10300^{x} (\ln (103^{302.2})) - 10300^{x} \ln (100^{302.2})}{100^{2 x}}$

Step 8.3.1.2

Factor $10300^{x}$ out of $- 10300^{x} \ln (100^{302.2})$ .

$\frac{10300^{x} (\ln (103^{302.2})) + 10300^{x} (- \ln (100^{302.2}))}{100^{2 x}}$

Step 8.3.1.3

Factor $10300^{x}$ out of $10300^{x} (\ln (103^{302.2})) + 10300^{x} (- \ln (100^{302.2}))$ .

$\frac{10300^{x} (\ln (103^{302.2}) - \ln (100^{302.2}))}{100^{2 x}}$

Step 8.3.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{10300^{x} \ln (\frac{103^{302.2}}{100^{302.2}})}{100^{2 x}}$