Calculus Examples

$x^{\frac{7}{2}} - 6 x^{2}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $x^{\frac{7}{2}} - 6 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{7}{2}}] + \frac{d}{d x} [- 6 x^{2}]$ .

$\frac{d}{d x} [x^{\frac{7}{2}}] + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [x^{\frac{7}{2}}]$ .

Tap for more steps...

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 $n = \frac{7}{2}$ .

$\frac{7}{2} x^{\frac{7}{2} - 1} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.2

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

$\frac{7}{2} x^{\frac{7}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.3

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

$\frac{7}{2} x^{\frac{7}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.4

Combine the numerators over the common denominator.

$\frac{7}{2} x^{\frac{7 - 1 \cdot 2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.5

Simplify the numerator.

Tap for more steps...

Step 1.1.2.5.1

Multiply $- 1$ by $2$ .

$\frac{7}{2} x^{\frac{7 - 2}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.2.5.2

Subtract $2$ from $7$ .

$\frac{7}{2} x^{\frac{5}{2}} + \frac{d}{d x} [- 6 x^{2}]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$\frac{7}{2} x^{\frac{5}{2}} - 6 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$\frac{7}{2} x^{\frac{5}{2}} - 6 (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 6$ .

$\frac{7}{2} x^{\frac{5}{2}} - 12 x$

Step 1.1.4

Combine $\frac{7}{2}$ and $x^{\frac{5}{2}}$ .

$f' (x) = \frac{7 x^{\frac{5}{2}}}{2} - 12 x$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{7 x^{\frac{5}{2}}}{2} - 12 x$ .

$\frac{7 x^{\frac{5}{2}}}{2} - 12 x$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{7 x^{\frac{5}{2}}}{2} - 12 x = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$\frac{7 x^{\frac{5}{2}}}{2} - 12 x = 0$

Step 2.2

Multiply each term in $\frac{7 x^{\frac{5}{2}}}{2} - 12 x = 0$ by $2$ to eliminate the fractions.

Tap for more steps...

Step 2.2.1

Multiply each term in $\frac{7 x^{\frac{5}{2}}}{2} - 12 x = 0$ by $2$ .

$\frac{7 x^{\frac{5}{2}}}{2} \cdot 2 - 12 x \cdot 2 = 0 \cdot 2$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.1.1.1

Cancel the common factor.

$\frac{7 x^{\frac{5}{2}}}{2} \cdot 2 - 12 x \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.1.2

Rewrite the expression.

$7 x^{\frac{5}{2}} - 12 x \cdot 2 = 0 \cdot 2$

Step 2.2.2.1.2

Multiply $2$ by $- 12$ .

$7 x^{\frac{5}{2}} - 24 x = 0 \cdot 2$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Multiply $0$ by $2$ .

$7 x^{\frac{5}{2}} - 24 x = 0$

Step 2.3

Factor $x$ out of $7 x^{\frac{5}{2}} - 24 x$ .

Tap for more steps...

Step 2.3.1

Factor $x$ out of $7 x^{\frac{5}{2}}$ .

$x (7 x^{\frac{3}{2}}) - 24 x = 0$

Step 2.3.2

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

$x (7 x^{\frac{3}{2}}) + x \cdot - 24 = 0$

Step 2.3.3

Factor $x$ out of $x (7 x^{\frac{3}{2}}) + x \cdot - 24$ .

$x (7 x^{\frac{3}{2}} - 24) = 0$

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 $0$ .

$x = 0$

$7 x^{\frac{3}{2}} - 24 = 0$

Step 2.5

Set $x$ equal to $0$ .

$x = 0$

Step 2.6

Set $7 x^{\frac{3}{2}} - 24$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.6.1

Set $7 x^{\frac{3}{2}} - 24$ equal to $0$ .

$7 x^{\frac{3}{2}} - 24 = 0$

Step 2.6.2

Solve $7 x^{\frac{3}{2}} - 24 = 0$ for $x$ .

Tap for more steps...

Step 2.6.2.1

Add $24$ to both sides of the equation.

$7 x^{\frac{3}{2}} = 24$

Step 2.6.2.2

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(7 x^{\frac{3}{2}})}^{\frac{2}{3}} = 24^{\frac{2}{3}}$

Step 2.6.2.3

Simplify the left side.

Tap for more steps...

Step 2.6.2.3.1

Simplify ${(7 x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Tap for more steps...

Step 2.6.2.3.1.1

Apply the product rule to $7 x^{\frac{3}{2}}$ .

$7^{\frac{2}{3}} {(x^{\frac{3}{2}})}^{\frac{2}{3}} = 24^{\frac{2}{3}}$

Step 2.6.2.3.1.2

Multiply the exponents in ${(x^{\frac{3}{2}})}^{\frac{2}{3}}$ .

Tap for more steps...

Step 2.6.2.3.1.2.1

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

$7^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 24^{\frac{2}{3}}$

Step 2.6.2.3.1.2.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.6.2.3.1.2.2.1

Cancel the common factor.

$7^{\frac{2}{3}} x^{\frac{3}{2} \cdot \frac{2}{3}} = 24^{\frac{2}{3}}$

Step 2.6.2.3.1.2.2.2

Rewrite the expression.

$7^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 24^{\frac{2}{3}}$

Step 2.6.2.3.1.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.2.3.1.2.3.1

Cancel the common factor.

$7^{\frac{2}{3}} x^{\frac{1}{2} \cdot 2} = 24^{\frac{2}{3}}$

Step 2.6.2.3.1.2.3.2

Rewrite the expression.

$7^{\frac{2}{3}} x^{1} = 24^{\frac{2}{3}}$

Step 2.6.2.3.1.3

Simplify.

$7^{\frac{2}{3}} x = 24^{\frac{2}{3}}$

Step 2.6.2.3.1.4

Reorder factors in $7^{\frac{2}{3}} x$ .

$x \cdot 7^{\frac{2}{3}} = 24^{\frac{2}{3}}$

Step 2.6.2.4

Divide each term in $x \cdot 7^{\frac{2}{3}} = 24^{\frac{2}{3}}$ by $7^{\frac{2}{3}}$ and simplify.

Tap for more steps...

Step 2.6.2.4.1

Divide each term in $x \cdot 7^{\frac{2}{3}} = 24^{\frac{2}{3}}$ by $7^{\frac{2}{3}}$ .

$\frac{x \cdot 7^{\frac{2}{3}}}{7^{\frac{2}{3}}} = \frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$

Step 2.6.2.4.2

Simplify the left side.

Tap for more steps...

Step 2.6.2.4.2.1

Cancel the common factor.

$\frac{x \cdot 7^{\frac{2}{3}}}{7^{\frac{2}{3}}} = \frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$

Step 2.6.2.4.2.2

Divide $x$ by $1$ .

$x = \frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$

Step 2.7

The final solution is all the values that make $x (7 x^{\frac{3}{2}} - 24) = 0$ true.

$x = 0, \frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$

Step 3

Find the values where the derivative is undefined.

Tap for more steps...

Step 3.1

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

$\frac{7 \sqrt{x^{5}}}{2} - 12 x$

Step 3.2

Set the radicand in $\sqrt{x^{5}}$ less than $0$ to find where the expression is undefined.

$x^{5} < 0$

Step 3.3

Solve for $x$ .

Tap for more steps...

Step 3.3.1

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

$\sqrt[5]{x^{5}} < \sqrt[5]{0}$

Step 3.3.2

Simplify the equation.

Tap for more steps...

Step 3.3.2.1

Simplify the left side.

Tap for more steps...

Step 3.3.2.1.1

Pull terms out from under the radical.

$x < \sqrt[5]{0}$

Step 3.3.2.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.2.1

Simplify $\sqrt[5]{0}$ .

Tap for more steps...

Step 3.3.2.2.1.1

Rewrite $0$ as $0^{5}$ .

$x < \sqrt[5]{0^{5}}$

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 $x^{\frac{7}{2}} - 6 x^{2}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 4.1

Evaluate at $x = 0$ .

Tap for more steps...

Step 4.1.1

Substitute $0$ for $x$ .

${(0)}^{\frac{7}{2}} - 6 {(0)}^{2}$

Step 4.1.2

Simplify.

Tap for more steps...

Step 4.1.2.1

Simplify each term.

Tap for more steps...

Step 4.1.2.1.1

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

${(0^{2})}^{\frac{7}{2}} - 6 {(0)}^{2}$

Step 4.1.2.1.2

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

$0^{2 (\frac{7}{2})} - 6 {(0)}^{2}$

Step 4.1.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.2.1.3.1

Cancel the common factor.

$0^{2 (\frac{7}{2})} - 6 {(0)}^{2}$

Step 4.1.2.1.3.2

Rewrite the expression.

$0^{7} - 6 {(0)}^{2}$

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 $x = \frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$ .

Tap for more steps...

Step 4.2.1

Substitute $\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$ for $x$ .

${(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{\frac{7}{2}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Simplify each term.

Tap for more steps...

Step 4.2.2.1.1

Apply the product rule to $\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$ .

$\frac{{(24^{\frac{2}{3}})}^{\frac{7}{2}}}{{(7^{\frac{2}{3}})}^{\frac{7}{2}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.2

Multiply the exponents in ${(24^{\frac{2}{3}})}^{\frac{7}{2}}$ .

Tap for more steps...

Step 4.2.2.1.2.1

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

$\frac{24^{\frac{2}{3} \cdot \frac{7}{2}}}{{(7^{\frac{2}{3}})}^{\frac{7}{2}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.2.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1.2.2.1

Cancel the common factor.

$\frac{24^{\frac{2}{3} \cdot \frac{7}{2}}}{{(7^{\frac{2}{3}})}^{\frac{7}{2}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.2.2.2

Rewrite the expression.

$\frac{24^{\frac{1}{3} \cdot 7}}{{(7^{\frac{2}{3}})}^{\frac{7}{2}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.2.3

Combine $\frac{1}{3}$ and $7$ .

$\frac{24^{\frac{7}{3}}}{{(7^{\frac{2}{3}})}^{\frac{7}{2}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.3

Multiply the exponents in ${(7^{\frac{2}{3}})}^{\frac{7}{2}}$ .

Tap for more steps...

Step 4.2.2.1.3.1

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

$\frac{24^{\frac{7}{3}}}{7^{\frac{2}{3} \cdot \frac{7}{2}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.3.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.2.1.3.2.1

Cancel the common factor.

$\frac{24^{\frac{7}{3}}}{7^{\frac{2}{3} \cdot \frac{7}{2}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.3.2.2

Rewrite the expression.

$\frac{24^{\frac{7}{3}}}{7^{\frac{1}{3} \cdot 7}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.3.3

Combine $\frac{1}{3}$ and $7$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 {(\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}})}^{2}$

Step 4.2.2.1.4

Apply the product rule to $\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 \frac{{(24^{\frac{2}{3}})}^{2}}{{(7^{\frac{2}{3}})}^{2}}$

Step 4.2.2.1.5

Multiply the exponents in ${(24^{\frac{2}{3}})}^{2}$ .

Tap for more steps...

Step 4.2.2.1.5.1

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

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 \frac{24^{\frac{2}{3} \cdot 2}}{{(7^{\frac{2}{3}})}^{2}}$

Step 4.2.2.1.5.2

Multiply $\frac{2}{3} \cdot 2$ .

Tap for more steps...

Step 4.2.2.1.5.2.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 \frac{24^{\frac{2 \cdot 2}{3}}}{{(7^{\frac{2}{3}})}^{2}}$

Step 4.2.2.1.5.2.2

Multiply $2$ by $2$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 \frac{24^{\frac{4}{3}}}{{(7^{\frac{2}{3}})}^{2}}$

Step 4.2.2.1.6

Multiply the exponents in ${(7^{\frac{2}{3}})}^{2}$ .

Tap for more steps...

Step 4.2.2.1.6.1

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

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 \frac{24^{\frac{4}{3}}}{7^{\frac{2}{3} \cdot 2}}$

Step 4.2.2.1.6.2

Multiply $\frac{2}{3} \cdot 2$ .

Tap for more steps...

Step 4.2.2.1.6.2.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 \frac{24^{\frac{4}{3}}}{7^{\frac{2 \cdot 2}{3}}}$

Step 4.2.2.1.6.2.2

Multiply $2$ by $2$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - 6 \frac{24^{\frac{4}{3}}}{7^{\frac{4}{3}}}$

Step 4.2.2.1.7

Combine $- 6$ and $\frac{24^{\frac{4}{3}}}{7^{\frac{4}{3}}}$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} + \frac{- 6 \cdot 24^{\frac{4}{3}}}{7^{\frac{4}{3}}}$

Step 4.2.2.1.8

Move the negative in front of the fraction.

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - \frac{6 \cdot 24^{\frac{4}{3}}}{7^{\frac{4}{3}}}$

Step 4.2.2.2

To write $- \frac{6 \cdot 24^{\frac{4}{3}}}{7^{\frac{4}{3}}}$ as a fraction with a common denominator, multiply by $\frac{7^{\frac{3}{3}}}{7^{\frac{3}{3}}}$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - \frac{6 \cdot 24^{\frac{4}{3}}}{7^{\frac{4}{3}}} \cdot \frac{7^{\frac{3}{3}}}{7^{\frac{3}{3}}}$

Step 4.2.2.3

Write each expression with a common denominator of $7^{\frac{7}{3}}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.2.2.3.1

Multiply $\frac{6 \cdot 24^{\frac{4}{3}}}{7^{\frac{4}{3}}}$ by $\frac{7^{\frac{3}{3}}}{7^{\frac{3}{3}}}$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - \frac{6 \cdot 24^{\frac{4}{3}} \cdot 7^{\frac{3}{3}}}{7^{\frac{4}{3}} \cdot 7^{\frac{3}{3}}}$

Step 4.2.2.3.2

Multiply $7^{\frac{4}{3}}$ by $7^{\frac{3}{3}}$ by adding the exponents.

Tap for more steps...

Step 4.2.2.3.2.1

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

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - \frac{6 \cdot 24^{\frac{4}{3}} \cdot 7^{\frac{3}{3}}}{7^{\frac{4}{3} + \frac{3}{3}}}$

Step 4.2.2.3.2.2

Combine the numerators over the common denominator.

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - \frac{6 \cdot 24^{\frac{4}{3}} \cdot 7^{\frac{3}{3}}}{7^{\frac{4 + 3}{3}}}$

Step 4.2.2.3.2.3

Add $4$ and $3$ .

$\frac{24^{\frac{7}{3}}}{7^{\frac{7}{3}}} - \frac{6 \cdot 24^{\frac{4}{3}} \cdot 7^{\frac{3}{3}}}{7^{\frac{7}{3}}}$

Step 4.2.2.4

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 4.2.2.4.1

Combine the numerators over the common denominator.

$\frac{24^{\frac{7}{3}} - 6 \cdot 24^{\frac{4}{3}} \cdot 7^{\frac{3}{3}}}{7^{\frac{7}{3}}}$

Step 4.2.2.4.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.2.4.2.1

Cancel the common factor.

$\frac{24^{\frac{7}{3}} - 6 \cdot 24^{\frac{4}{3}} \cdot 7^{\frac{3}{3}}}{7^{\frac{7}{3}}}$

Step 4.2.2.4.2.2

Rewrite the expression.

$\frac{24^{\frac{7}{3}} - 6 \cdot 24^{\frac{4}{3}} \cdot 7^{1}}{7^{\frac{7}{3}}}$

Step 4.2.2.5

Simplify the numerator.

Tap for more steps...

Step 4.2.2.5.1

Evaluate the exponent.

$\frac{24^{\frac{7}{3}} - 6 \cdot 24^{\frac{4}{3}} \cdot 7}{7^{\frac{7}{3}}}$

Step 4.2.2.5.2

Multiply $7$ by $- 6$ .

$\frac{24^{\frac{7}{3}} - 42 \cdot 24^{\frac{4}{3}}}{7^{\frac{7}{3}}}$

Step 4.3

List all of the points.

$(0, 0), (\frac{24^{\frac{2}{3}}}{7^{\frac{2}{3}}}, \frac{24^{\frac{7}{3}} - 42 \cdot 24^{\frac{4}{3}}}{7^{\frac{7}{3}}})$

Step 5