Calculus Examples

$f (x) = 7.2956 \ln (0.0645012 x^{0.95} + 1) (200)$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Multiply $200$ by $7.2956$ .

$\frac{d}{d x} [1459.12 \ln (0.0645012 x^{0.95} + 1)]$

Step 1.2

Since $1459.12$ is constant with respect to $x$ , the derivative of $1459.12 \ln (0.0645012 x^{0.95} + 1)$ with respect to $x$ is $1459.12 \frac{d}{d x} [\ln (0.0645012 x^{0.95} + 1)]$ .

$1459.12 \frac{d}{d x} [\ln (0.0645012 x^{0.95} + 1)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = 0.0645012 x^{0.95} + 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $0.0645012 x^{0.95} + 1$ .

$1459.12 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [0.0645012 x^{0.95} + 1])$

Step 2.2

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

$1459.12 (\frac{1}{u} \frac{d}{d x} [0.0645012 x^{0.95} + 1])$

Step 2.3

Replace all occurrences of $u$ with $0.0645012 x^{0.95} + 1$ .

$1459.12 (\frac{1}{0.0645012 x^{0.95} + 1} \frac{d}{d x} [0.0645012 x^{0.95} + 1])$

Step 3

Differentiate.

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Step 3.1

Combine $\frac{1}{0.0645012 x^{0.95} + 1}$ and $1459.12$ .

$\frac{1459.12}{0.0645012 x^{0.95} + 1} \frac{d}{d x} [0.0645012 x^{0.95} + 1]$

Step 3.2

By the Sum Rule, the derivative of $0.0645012 x^{0.95} + 1$ with respect to $x$ is $\frac{d}{d x} [0.0645012 x^{0.95}] + \frac{d}{d x} [1]$ .

$\frac{1459.12}{0.0645012 x^{0.95} + 1} (\frac{d}{d x} [0.0645012 x^{0.95}] + \frac{d}{d x} [1])$

Step 3.3

Since $0.0645012$ is constant with respect to $x$ , the derivative of $0.0645012 x^{0.95}$ with respect to $x$ is $0.0645012 \frac{d}{d x} [x^{0.95}]$ .

$\frac{1459.12}{0.0645012 x^{0.95} + 1} (0.0645012 \frac{d}{d x} [x^{0.95}] + \frac{d}{d x} [1])$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 0.95$ .

$\frac{1459.12}{0.0645012 x^{0.95} + 1} (0.0645012 (0.95 x^{- 0.05}) + \frac{d}{d x} [1])$

Step 3.5

Multiply $0.95$ by $0.0645012$ .

$\frac{1459.12}{0.0645012 x^{0.95} + 1} (0.06127614 x^{- 0.05} + \frac{d}{d x} [1])$

Step 3.6

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1459.12}{0.0645012 x^{0.95} + 1} (0.06127614 x^{- 0.05} + 0)$

Step 3.7

Combine fractions.

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Step 3.7.1

Add $0.06127614 x^{- 0.05}$ and $0$ .

$\frac{1459.12}{0.0645012 x^{0.95} + 1} (0.06127614 x^{- 0.05})$

Step 3.7.2

Combine $0.06127614$ and $\frac{1459.12}{0.0645012 x^{0.95} + 1}$ .

$\frac{0.06127614 \cdot 1459.12}{0.0645012 x^{0.95} + 1} x^{- 0.05}$

Step 3.7.3

Multiply $0.06127614$ by $1459.12$ .

$\frac{89.40924139}{0.0645012 x^{0.95} + 1} x^{- 0.05}$

Step 3.7.4

Combine $\frac{89.40924139}{0.0645012 x^{0.95} + 1}$ and $x^{- 0.05}$ .

$\frac{89.40924139 x^{- 0.05}}{0.0645012 x^{0.95} + 1}$

Step 3.7.5

Move $x^{- 0.05}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{89.40924139}{(0.0645012 x^{0.95} + 1) x^{0.05}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{89.40924139}{0.0645012 x^{0.95} x^{0.05} + 1 x^{0.05}}$

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $x^{0.95}$ by $x^{0.05}$ by adding the exponents.

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Step 4.2.1.1

Move $x^{0.05}$ .

$\frac{89.40924139}{0.0645012 (x^{0.05} x^{0.95}) + 1 x^{0.05}}$

Step 4.2.1.2

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

$\frac{89.40924139}{0.0645012 x^{0.05 + 0.95} + 1 x^{0.05}}$

Step 4.2.1.3

Add $0.05$ and $0.95$ .

$\frac{89.40924139}{0.0645012 x^{1} + 1 x^{0.05}}$

Step 4.2.2

Simplify $0.0645012 x^{1}$ .

$\frac{89.40924139}{0.0645012 x + 1 x^{0.05}}$

Step 4.2.3

Multiply $x^{0.05}$ by $1$ .

$\frac{89.40924139}{0.0645012 x + x^{0.05}}$