Solve 6x^frac{2{3}}*3x*x^-3

$$6 x ^ { \frac { 2 } { 3 } } \cdot 3 x \cdot x ^ { - 3 }$$

Evaluate

$$\frac{18}{x^{\frac{4}{3}}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and 1 to get \frac{5}{3}.

$$6x^{\frac{5}{3}}\times 3x^{-3}$$

To multiply powers of the same base, add their exponents. Add \frac{5}{3} and -3 to get -\frac{4}{3}.

$$6x^{-\frac{4}{3}}\times 3$$

Multiply 6 and 3 to get 18.

$$18x^{-\frac{4}{3}}$$

Differentiate w.r.t. x

$$-\frac{24}{x^{\frac{7}{3}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{2}{3} and 1 to get \frac{5}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{\frac{5}{3}}\times 3x^{-3})$$

To multiply powers of the same base, add their exponents. Add \frac{5}{3} and -3 to get -\frac{4}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{-\frac{4}{3}}\times 3)$$

Multiply 6 and 3 to get 18.

$$\frac{\mathrm{d}}{\mathrm{d}x}(18x^{-\frac{4}{3}})$$

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

$$-\frac{4}{3}\times 18x^{-\frac{4}{3}-1}$$

Multiply -\frac{4}{3} times 18.

$$-24x^{-\frac{4}{3}-1}$$

Subtract 1 from -\frac{4}{3}.

$$-24x^{-\frac{7}{3}}$$