Solve (144k^frac{4{11}})^frac{1{2}}(k^frac{-5{22}})

$$( 144 k ^ { \frac { 4 } { 11 } } ) ^ { \frac { 1 } { 2 } } ( k ^ { \frac { - 5 } { 22 } } )$$

Evaluate

$$\frac{12}{\sqrt[22]{k}}$$

Solution Steps

Expand \left(144k^{\frac{4}{11}}\right)^{\frac{1}{2}}.

$$144^{\frac{1}{2}}\left(k^{\frac{4}{11}}\right)^{\frac{1}{2}}k^{\frac{-5}{22}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{4}{11} and \frac{1}{2} to get \frac{2}{11}.

$$144^{\frac{1}{2}}k^{\frac{2}{11}}k^{\frac{-5}{22}}$$

Calculate 144 to the power of \frac{1}{2} and get 12.

$$12k^{\frac{2}{11}}k^{\frac{-5}{22}}$$

Fraction \frac{-5}{22} can be rewritten as -\frac{5}{22} by extracting the negative sign.

$$12k^{\frac{2}{11}}k^{-\frac{5}{22}}$$

To multiply powers of the same base, add their exponents. Add \frac{2}{11} and -\frac{5}{22} to get -\frac{1}{22}.

$$12k^{-\frac{1}{22}}$$

Differentiate w.r.t. k

$$-\frac{6}{11k^{\frac{23}{22}}}$$

Steps Using Definition of a Derivative

Expand \left(144k^{\frac{4}{11}}\right)^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(144^{\frac{1}{2}}\left(k^{\frac{4}{11}}\right)^{\frac{1}{2}}k^{\frac{-5}{22}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{4}{11} and \frac{1}{2} to get \frac{2}{11}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(144^{\frac{1}{2}}k^{\frac{2}{11}}k^{\frac{-5}{22}})$$

Calculate 144 to the power of \frac{1}{2} and get 12.

$$\frac{\mathrm{d}}{\mathrm{d}k}(12k^{\frac{2}{11}}k^{\frac{-5}{22}})$$

Fraction \frac{-5}{22} can be rewritten as -\frac{5}{22} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}k}(12k^{\frac{2}{11}}k^{-\frac{5}{22}})$$

To multiply powers of the same base, add their exponents. Add \frac{2}{11} and -\frac{5}{22} to get -\frac{1}{22}.

$$\frac{\mathrm{d}}{\mathrm{d}k}(12k^{-\frac{1}{22}})$$

The derivative of ax^{n} is nax^{n-1}.

$$-\frac{1}{22}\times 12k^{-\frac{1}{22}-1}$$

Multiply -\frac{1}{22} times 12.

$$-\frac{6}{11}k^{-\frac{1}{22}-1}$$

Subtract 1 from -\frac{1}{22}.

$$-\frac{6}{11}k^{-\frac{23}{22}}$$