Calculus Examples

$\frac{d π}{d x} = 200 x^{1.075}$

Step 1

Reduce the expression $\frac{d π}{d x}$ by cancelling the common factors.

Tap for more steps...

Step 1.1

Cancel the common factor.

$\frac{d π}{d x} = 200 x^{1.075}$

Step 1.2

Rewrite the expression.

$\frac{π}{x} = 200 x^{1.075}$

Step 2

Solve the equation.

Tap for more steps...

Step 2.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 2.1.2

The LCM of one and any expression is the expression.

$x$

Step 2.2

Multiply each term in $\frac{π}{x} = 200 x^{1.075}$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 2.2.1

Multiply each term in $\frac{π}{x} = 200 x^{1.075}$ by $x$ .

$\frac{π}{x} x = 200 x^{1.075} x$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor.

$\frac{π}{x} x = 200 x^{1.075} x$

Step 2.2.2.1.2

Rewrite the expression.

$π = 200 x^{1.075} x$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Multiply $x^{1.075}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.2.3.1.1

Move $x$ .

$π = 200 (x \cdot x^{1.075})$

Step 2.2.3.1.2

Multiply $x$ by $x^{1.075}$ .

Tap for more steps...

Step 2.2.3.1.2.1

Raise $x$ to the power of $1$ .

$π = 200 (x^{1} x^{1.075})$

Step 2.2.3.1.2.2

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

$π = 200 x^{1 + 1.075}$

Step 2.2.3.1.3

Add $1$ and $1.075$ .

$π = 200 x^{2.075}$

Step 2.3

Solve the equation.

Tap for more steps...

Step 2.3.1

Rewrite the equation as $200 x^{2.075} = π$ .

$200 x^{2.075} = π$

Step 2.3.2

Divide each term in $200 x^{2.075} = π$ by $200$ and simplify.

Tap for more steps...

Step 2.3.2.1

Divide each term in $200 x^{2.075} = π$ by $200$ .

$\frac{200 x^{2.075}}{200} = \frac{π}{200}$

Step 2.3.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.1

Cancel the common factor of $200$ .

Tap for more steps...

Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{200 x^{2.075}}{200} = \frac{π}{200}$

Step 2.3.2.2.1.2

Divide $x^{2.075}$ by $1$ .

$x^{2.075} = \frac{π}{200}$

Step 2.3.3

Convert the decimal exponent to a fractional exponent.

Tap for more steps...

Step 2.3.3.1

Convert the decimal number to a fraction by placing the decimal number over a power of ten. Since there are $3$ numbers to the right of the decimal point, place the decimal number over $10^{3}$ $(1000)$ . Next, add the whole number to the left of the decimal.

$x^{2 \frac{75}{1000}} = \frac{π}{200}$

Step 2.3.3.2

Reduce the fractional part of the mixed number.

$x^{2 \frac{3}{40}} = \frac{π}{200}$

Step 2.3.3.3

Convert $2 \frac{3}{40}$ to an improper fraction.

Tap for more steps...

Step 2.3.3.3.1

A mixed number is an addition of its whole and fractional parts.

$x^{2 + \frac{3}{40}} = \frac{π}{200}$

Step 2.3.3.3.2

Add $2$ and $\frac{3}{40}$ .

Tap for more steps...

Step 2.3.3.3.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{40}{40}$ .

$x^{2 \cdot \frac{40}{40} + \frac{3}{40}} = \frac{π}{200}$

Step 2.3.3.3.2.2

Combine $2$ and $\frac{40}{40}$ .

$x^{\frac{2 \cdot 40}{40} + \frac{3}{40}} = \frac{π}{200}$

Step 2.3.3.3.2.3

Combine the numerators over the common denominator.

$x^{\frac{2 \cdot 40 + 3}{40}} = \frac{π}{200}$

Step 2.3.3.3.2.4

Simplify the numerator.

Tap for more steps...

Step 2.3.3.3.2.4.1

Multiply $2$ by $40$ .

$x^{\frac{80 + 3}{40}} = \frac{π}{200}$

Step 2.3.3.3.2.4.2

Add $80$ and $3$ .

$x^{\frac{83}{40}} = \frac{π}{200}$

Step 2.3.4

Raise each side of the equation to the power of $\frac{1}{2.075}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{83}{40}})}^{\frac{1}{2.075}} = {(\frac{π}{200})}^{\frac{1}{2.075}}$

Step 2.3.5

Simplify the exponent.

Tap for more steps...

Step 2.3.5.1

Simplify the left side.

Tap for more steps...

Step 2.3.5.1.1

Simplify ${(x^{\frac{83}{40}})}^{\frac{1}{2.075}}$ .

Tap for more steps...

Step 2.3.5.1.1.1

Multiply the exponents in ${(x^{\frac{83}{40}})}^{\frac{1}{2.075}}$ .

Tap for more steps...

Step 2.3.5.1.1.1.1

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

$x^{\frac{83}{40} \cdot \frac{1}{2.075}} = {(\frac{π}{200})}^{\frac{1}{2.075}}$

Step 2.3.5.1.1.1.2

Cancel the common factor of $2.075$ .

Tap for more steps...

Step 2.3.5.1.1.1.2.1

Factor $2.075$ out of $83$ .

$x^{\frac{2.075 (40)}{40} \cdot \frac{1}{2.075}} = {(\frac{π}{200})}^{\frac{1}{2.075}}$

Step 2.3.5.1.1.1.2.2

Cancel the common factor.

$x^{\frac{2.075 \cdot 40}{40} \cdot \frac{1}{2.075}} = {(\frac{π}{200})}^{\frac{1}{2.075}}$

Step 2.3.5.1.1.1.2.3

Rewrite the expression.

$x^{\frac{40}{40}} = {(\frac{π}{200})}^{\frac{1}{2.075}}$

Step 2.3.5.1.1.1.3

Divide $40$ by $40$ .

$x^{1} = {(\frac{π}{200})}^{\frac{1}{2.075}}$

Step 2.3.5.1.1.2

Simplify.

$x = {(\frac{π}{200})}^{\frac{1}{2.075}}$

Step 2.3.5.2

Simplify the right side.

Tap for more steps...

Step 2.3.5.2.1

Simplify ${(\frac{π}{200})}^{\frac{1}{2.075}}$ .

Tap for more steps...

Step 2.3.5.2.1.1

Divide $1$ by $2.075$ .

$x = {(\frac{π}{200})}^{0.48192771}$

Step 2.3.5.2.1.2

Apply the product rule to $\frac{π}{200}$ .

$x = \frac{π^{0.48192771}}{200^{0.48192771}}$

Step 2.3.5.2.1.3

Raise $200$ to the power of $0.48192771$ .

$x = \frac{π^{0.48192771}}{12.85079863}$

Step 2.3.5.2.1.4

Replace $π$ with an approximation.

$x = \frac{{3.14159265}^{0.48192771}}{12.85079863}$

Step 2.3.5.2.1.5

Raise $3.14159265$ to the power of $0.48192771$ .

$x = \frac{1.73616221}{12.85079863}$

Step 2.3.5.2.1.6

Divide $1.73616221$ by $12.85079863$ .

$x = 0.1351015$