Calculus Examples

$x^{\frac{5}{2}} - 6 x^{2}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{\frac{5}{2}} - 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- 6 x^{2}]$ .

$\frac{d}{d x} [x^{\frac{5}{2}}] + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{5}{2}}]$ .

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Step 1.1.2.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{5}{2}$ .

$\frac{5}{2} x^{\frac{5}{2} - 1} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5}{2} x^{\frac{5}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.3

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{5}{2} x^{\frac{5}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.4

Combine the numerators over the common denominator.

$\frac{5}{2} x^{\frac{5 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $- 1$ by $2$ .

$\frac{5}{2} x^{\frac{5 - 2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.5.2

Subtract $2$ from $5$ .

$\frac{5}{2} x^{\frac{3}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 6 x^{2}]$ .

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Step 1.1.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$\frac{5}{2} x^{\frac{3}{2}} - 6 \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

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

$\frac{5}{2} x^{\frac{3}{2}} - 6 (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 6$ .

$\frac{5}{2} x^{\frac{3}{2}} - 12 x$

Step 1.1.4

Combine $\frac{5}{2}$ and $x^{\frac{3}{2}}$ .

$f' (x) = \frac{5 x^{\frac{3}{2}}}{2} - 12 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{5 x^{\frac{3}{2}}}{2} - 12 x$ .

$\frac{5 x^{\frac{3}{2}}}{2} - 12 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{5 x^{\frac{3}{2}}}{2} - 12 x = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{5 x^{\frac{3}{2}}}{2} - 12 x = 0$

Step 2.2

Multiply each term in $\frac{5 x^{\frac{3}{2}}}{2} - 12 x = 0$ by $2$ to eliminate the fractions.

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Step 2.2.1

Multiply each term in $\frac{5 x^{\frac{3}{2}}}{2} - 12 x = 0$ by $2$ .

$\frac{5 x^{\frac{3}{2}}}{2} \cdot 2 - 12 x \cdot 2 = 0 \cdot 2$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.1

Cancel the common factor.

$\frac{5 x^{\frac{3}{2}}}{2} \cdot 2 - 12 x \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.1.2

Rewrite the expression.

$5 x^{\frac{3}{2}} - 12 x \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.2

Multiply $2$ by $- 12$ .

$5 x^{\frac{3}{2}} - 24 x = 0 \cdot 2$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Multiply $0$ by $2$ .

$5 x^{\frac{3}{2}} - 24 x = 0$

Step 2.3

Factor $x$ out of $5 x^{\frac{3}{2}} - 24 x$ .

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Step 2.3.1

Factor $x$ out of $5 x^{\frac{3}{2}}$ .

$x (5 x^{\frac{1}{2}}) - 24 x = 0$

Step 2.3.2

Factor $x$ out of $- 24 x$ .

$x (5 x^{\frac{1}{2}}) + x \cdot - 24 = 0$

Step 2.3.3

Factor $x$ out of $x (5 x^{\frac{1}{2}}) + x \cdot - 24$ .

$x (5 x^{\frac{1}{2}} - 24) = 0$

Step 2.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$5 x^{\frac{1}{2}} - 24 = 0$

Step 2.5

Set $x$ equal to $0$ .

$x = 0$

Step 2.6

Set $5 x^{\frac{1}{2}} - 24$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $5 x^{\frac{1}{2}} - 24$ equal to $0$ .

$5 x^{\frac{1}{2}} - 24 = 0$

Step 2.6.2

Solve $5 x^{\frac{1}{2}} - 24 = 0$ for $x$ .

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Step 2.6.2.1

Add $24$ to both sides of the equation.

$5 x^{\frac{1}{2}} = 24$

Step 2.6.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(5 x^{\frac{1}{2}})}^{2} = 24^{2}$

Step 2.6.2.3

Simplify the exponent.

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Step 2.6.2.3.1

Simplify the left side.

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Step 2.6.2.3.1.1

Simplify ${(5 x^{\frac{1}{2}})}^{2}$ .

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Step 2.6.2.3.1.1.1

Apply the product rule to $5 x^{\frac{1}{2}}$ .

$5^{2} {(x^{\frac{1}{2}})}^{2} = 24^{2}$

Step 2.6.2.3.1.1.2

Raise $5$ to the power of $2$ .

$25 {(x^{\frac{1}{2}})}^{2} = 24^{2}$

Step 2.6.2.3.1.1.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.6.2.3.1.1.3.1

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

$25 x^{\frac{1}{2} \cdot 2} = 24^{2}$

Step 2.6.2.3.1.1.3.2

Cancel the common factor of $2$ .

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Step 2.6.2.3.1.1.3.2.1

Cancel the common factor.

$25 x^{\frac{1}{2} \cdot 2} = 24^{2}$

Step 2.6.2.3.1.1.3.2.2

Rewrite the expression.

$25 x^{1} = 24^{2}$

Step 2.6.2.3.1.1.4

Simplify.

$25 x = 24^{2}$

Step 2.6.2.3.2

Simplify the right side.

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Step 2.6.2.3.2.1

Raise $24$ to the power of $2$ .

$25 x = 576$

Step 2.6.2.4

Divide each term in $25 x = 576$ by $25$ and simplify.

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Step 2.6.2.4.1

Divide each term in $25 x = 576$ by $25$ .

$\frac{25 x}{25} = \frac{576}{25}$

Step 2.6.2.4.2

Simplify the left side.

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Step 2.6.2.4.2.1

Cancel the common factor of $25$ .

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Step 2.6.2.4.2.1.1

Cancel the common factor.

$\frac{25 x}{25} = \frac{576}{25}$

Step 2.6.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{576}{25}$

Step 2.7

The final solution is all the values that make $x (5 x^{\frac{1}{2}} - 24) = 0$ true.

$x = 0, \frac{576}{25}$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{5 \sqrt{x^{3}}}{2} - 12 x$

Step 3.2

Set the radicand in $\sqrt{x^{3}}$ less than $0$ to find where the expression is undefined.

$x^{3} < 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt[3]{x^{3}} < \sqrt[3]{0}$

Step 3.3.2

Simplify the equation.

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Step 3.3.2.1

Simplify the left side.

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Step 3.3.2.1.1

Pull terms out from under the radical.

$x < \sqrt[3]{0}$

Step 3.3.2.2

Simplify the right side.

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Step 3.3.2.2.1

Simplify $\sqrt[3]{0}$ .

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Step 3.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$x < \sqrt[3]{0^{3}}$

Step 3.3.2.2.1.2

Pull terms out from under the radical.

$x < 0$

Step 3.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x < 0$

$(- \infty, 0)$

$x < 0$

$(- \infty, 0)$

Step 4

Evaluate $x^{\frac{5}{2}} - 6 x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 0$ .

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Step 4.1.1

Substitute $0$ for $x$ .

${(0)}^{\frac{5}{2}} - 6 {(0)}^{2}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Rewrite $0$ as $0^{2}$ .

${(0^{2})}^{\frac{5}{2}} - 6 {(0)}^{2}$

Step 4.1.2.1.2

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

$0^{2 (\frac{5}{2})} - 6 {(0)}^{2}$

Step 4.1.2.1.3

Cancel the common factor of $2$ .

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Step 4.1.2.1.3.1

Cancel the common factor.

$0^{2 (\frac{5}{2})} - 6 {(0)}^{2}$

Step 4.1.2.1.3.2

Rewrite the expression.

$0^{5} - 6 {(0)}^{2}$

Step 4.1.2.1.4

Raising $0$ to any positive power yields $0$ .

$0 - 6 {(0)}^{2}$

Step 4.1.2.1.5

Raising $0$ to any positive power yields $0$ .

$0 - 6 \cdot 0$

Step 4.1.2.1.6

Multiply $- 6$ by $0$ .

$0 + 0$

Step 4.1.2.2

Add $0$ and $0$ .

$0$

Step 4.2

Evaluate at $x = \frac{576}{25}$ .

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Step 4.2.1

Substitute $\frac{576}{25}$ for $x$ .

${(\frac{576}{25})}^{\frac{5}{2}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Apply the product rule to $\frac{576}{25}$ .

$\frac{576^{\frac{5}{2}}}{25^{\frac{5}{2}}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.2

Simplify the numerator.

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Step 4.2.2.1.2.1

Rewrite $576$ as $24^{2}$ .

$\frac{{(24^{2})}^{\frac{5}{2}}}{25^{\frac{5}{2}}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.2.2

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

$\frac{24^{2 (\frac{5}{2})}}{25^{\frac{5}{2}}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.2.3

Cancel the common factor of $2$ .

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Step 4.2.2.1.2.3.1

Cancel the common factor.

$\frac{24^{2 (\frac{5}{2})}}{25^{\frac{5}{2}}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.2.3.2

Rewrite the expression.

$\frac{24^{5}}{25^{\frac{5}{2}}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.2.4

Raise $24$ to the power of $5$ .

$\frac{7962624}{25^{\frac{5}{2}}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.3

Simplify the denominator.

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Step 4.2.2.1.3.1

Rewrite $25$ as $5^{2}$ .

$\frac{7962624}{{(5^{2})}^{\frac{5}{2}}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.3.2

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

$\frac{7962624}{5^{2 (\frac{5}{2})}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 4.2.2.1.3.3.1

Cancel the common factor.

$\frac{7962624}{5^{2 (\frac{5}{2})}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.3.3.2

Rewrite the expression.

$\frac{7962624}{5^{5}} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.3.4

Raise $5$ to the power of $5$ .

$\frac{7962624}{3125} - 6 {(\frac{576}{25})}^{2}$

Step 4.2.2.1.4

Apply the product rule to $\frac{576}{25}$ .

$\frac{7962624}{3125} - 6 \frac{576^{2}}{25^{2}}$

Step 4.2.2.1.5

Raise $576$ to the power of $2$ .

$\frac{7962624}{3125} - 6 \frac{331776}{25^{2}}$

Step 4.2.2.1.6

Raise $25$ to the power of $2$ .

$\frac{7962624}{3125} - 6 (\frac{331776}{625})$

Step 4.2.2.1.7

Multiply $- 6 (\frac{331776}{625})$ .

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Step 4.2.2.1.7.1

Combine $- 6$ and $\frac{331776}{625}$ .

$\frac{7962624}{3125} + \frac{- 6 \cdot 331776}{625}$

Step 4.2.2.1.7.2

Multiply $- 6$ by $331776$ .

$\frac{7962624}{3125} + \frac{- 1990656}{625}$

Step 4.2.2.1.8

Move the negative in front of the fraction.

$\frac{7962624}{3125} - \frac{1990656}{625}$

Step 4.2.2.2

To write $- \frac{1990656}{625}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{7962624}{3125} - \frac{1990656}{625} \cdot \frac{5}{5}$

Step 4.2.2.3

Write each expression with a common denominator of $3125$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.2.3.1

Multiply $\frac{1990656}{625}$ by $\frac{5}{5}$ .

$\frac{7962624}{3125} - \frac{1990656 \cdot 5}{625 \cdot 5}$

Step 4.2.2.3.2

Multiply $625$ by $5$ .

$\frac{7962624}{3125} - \frac{1990656 \cdot 5}{3125}$

Step 4.2.2.4

Combine the numerators over the common denominator.

$\frac{7962624 - 1990656 \cdot 5}{3125}$

Step 4.2.2.5

Simplify the numerator.

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Step 4.2.2.5.1

Multiply $- 1990656$ by $5$ .

$\frac{7962624 - 9953280}{3125}$

Step 4.2.2.5.2

Subtract $9953280$ from $7962624$ .

$\frac{- 1990656}{3125}$

Step 4.2.2.6

Move the negative in front of the fraction.

$- \frac{1990656}{3125}$

Step 4.3

List all of the points.

$(0, 0), (\frac{576}{25}, - \frac{1990656}{3125})$

Step 5