Solve 6n^3/n^frac{1{12}*n^frac{1{4}}}

$$\frac { 6 n ^ { 3 } } { n ^ { \frac { 1 } { 12 } } \cdot n ^ { \frac { 1 } { 4 } } }$$

Evaluate

$$6n^{\frac{8}{3}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{12} and \frac{1}{4} to get \frac{1}{3}.

$$\frac{6n^{3}}{n^{\frac{1}{3}}}$$

Cancel out \sqrt[3]{n} in both numerator and denominator.

$$6n^{\frac{8}{3}}$$

Differentiate w.r.t. n

$$16n^{\frac{5}{3}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{12} and \frac{1}{4} to get \frac{1}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}n}(\frac{6n^{3}}{n^{\frac{1}{3}}})$$

Cancel out \sqrt[3]{n} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}n}(6n^{\frac{8}{3}})$$

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

$$\frac{8}{3}\times 6n^{\frac{8}{3}-1}$$

Multiply \frac{8}{3} times 6.

$$16n^{\frac{8}{3}-1}$$

Subtract 1 from \frac{8}{3}.

$$16n^{\frac{5}{3}}$$