Calculus Examples

$f (x) = \ln (30 x + \frac{85 (y^{2})}{569 x^{3.63} x^{6}})$

Step 1

Multiply $x^{3.63}$ by $x^{6}$ by adding the exponents.

Tap for more steps...

Step 1.1

Move $x^{6}$ .

$\frac{d}{d x} [\ln (30 x + \frac{85 y^{2}}{569 (x^{6} x^{3.63})})]$

Step 1.2

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

$\frac{d}{d x} [\ln (30 x + \frac{85 y^{2}}{569 x^{6 + 3.63}})]$

Step 1.3

Add $6$ and $3.63$ .

$\frac{d}{d x} [\ln (30 x + \frac{85 y^{2}}{569 x^{9.63}})]$

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 $g (x) = 30 x + \frac{85 y^{2}}{569 x^{9.63}}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u$ as $30 x + \frac{85 y^{2}}{569 x^{9.63}}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [30 x + \frac{85 y^{2}}{569 x^{9.63}}]$

Step 2.2

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

$\frac{1}{u} \frac{d}{d x} [30 x + \frac{85 y^{2}}{569 x^{9.63}}]$

Step 2.3

Replace all occurrences of $u$ with $30 x + \frac{85 y^{2}}{569 x^{9.63}}$ .

Tap for more steps...

Step 2.3.1

Move $x^{6}$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 (x^{6} x^{3.63})}} \frac{d}{d x} [30 x + \frac{85 y^{2}}{569 x^{9.63}}]$

Step 2.3.2

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

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{6 + 3.63}}} \frac{d}{d x} [30 x + \frac{85 y^{2}}{569 x^{9.63}}]$

Step 2.3.3

Add $6$ and $3.63$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} \frac{d}{d x} [30 x + \frac{85 y^{2}}{569 x^{9.63}}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $30 x + \frac{85 y^{2}}{569 x^{9.63}}$ with respect to $x$ is $\frac{d}{d x} [30 x] + \frac{d}{d x} [\frac{85 y^{2}}{569 x^{9.63}}]$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (\frac{d}{d x} [30 x] + \frac{d}{d x} [\frac{85 y^{2}}{569 x^{9.63}}])$

Step 3.2

Since $30$ is constant with respect to $x$ , the derivative of $30 x$ with respect to $x$ is $30 \frac{d}{d x} [x]$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{85 y^{2}}{569 x^{9.63}}])$

Step 3.3

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

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 \cdot 1 + \frac{d}{d x} [\frac{85 y^{2}}{569 x^{9.63}}])$

Step 3.4

Multiply $30$ by $1$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{d}{d x} [\frac{85 y^{2}}{569 x^{9.63}}])$

Step 3.5

Since $\frac{85 y^{2}}{569}$ is constant with respect to $x$ , the derivative of $\frac{85 y^{2}}{569 x^{9.63}}$ with respect to $x$ is $\frac{85 y^{2}}{569} \frac{d}{d x} [\frac{1}{x^{9.63}}]$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{85 y^{2}}{569} \frac{d}{d x} [\frac{1}{x^{9.63}}])$

Step 3.6

Apply basic rules of exponents.

Tap for more steps...

Step 3.6.1

Rewrite $\frac{1}{x^{9.63}}$ as ${(x^{9.63})}^{- 1}$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{85 y^{2}}{569} \frac{d}{d x} [{(x^{9.63})}^{- 1}])$

Step 3.6.2

Multiply the exponents in ${(x^{9.63})}^{- 1}$ .

Tap for more steps...

Step 3.6.2.1

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

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{85 y^{2}}{569} \frac{d}{d x} [x^{9.63 \cdot - 1}])$

Step 3.6.2.2

Multiply $9.63$ by $- 1$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{85 y^{2}}{569} \frac{d}{d x} [x^{- 9.63}])$

Step 3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 9.63$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{85 y^{2}}{569} (- 9.63 x^{- 10.63}))$

Step 3.8

Combine fractions.

Tap for more steps...

Step 3.8.1

Combine $- 9.63$ and $\frac{85 y^{2}}{569}$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{- 9.63 (85 y^{2})}{569} x^{- 10.63})$

Step 3.8.2

Multiply $85$ by $- 9.63$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{- 818.55 y^{2}}{569} x^{- 10.63})$

Step 3.8.3

Combine $\frac{- 818.55 y^{2}}{569}$ and $x^{- 10.63}$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{- 818.55 y^{2} x^{- 10.63}}{569})$

Step 3.8.4

Simplify the expression.

Tap for more steps...

Step 3.8.4.1

Move $x^{- 10.63}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 + \frac{- 818.55 y^{2}}{569 x^{10.63}})$

Step 3.8.4.2

Move the negative in front of the fraction.

$\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 - \frac{818.55 y^{2}}{569 x^{10.63}})$

Step 3.8.4.3

Reorder the factors of $\frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}} (30 - \frac{818.55 y^{2}}{569 x^{10.63}})$ .

$(30 - \frac{818.55 y^{2}}{569 x^{10.63}}) \frac{1}{30 x + \frac{85 y^{2}}{569 x^{9.63}}}$