Solve x^frac{1{2}}*2x^frac{4{3}}*4x^frac{5{3}}

$$x ^ { \frac { 1 } { 2 } } \cdot 2 x ^ { \frac { 4 } { 3 } } \cdot 4 x ^ { \frac { 5 } { 3 } }$$

Evaluate

$$8x^{\frac{7}{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{4}{3} to get \frac{11}{6}.

$$x^{\frac{11}{6}}\times 2\times 4x^{\frac{5}{3}}$$

To multiply powers of the same base, add their exponents. Add \frac{11}{6} and \frac{5}{3} to get \frac{7}{2}.

$$x^{\frac{7}{2}}\times 2\times 4$$

Multiply 2 and 4 to get 8.

$$x^{\frac{7}{2}}\times 8$$

Differentiate w.r.t. x

$$28x^{\frac{5}{2}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{4}{3} to get \frac{11}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{11}{6}}\times 2\times 4x^{\frac{5}{3}})$$

To multiply powers of the same base, add their exponents. Add \frac{11}{6} and \frac{5}{3} to get \frac{7}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{7}{2}}\times 2\times 4)$$

Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{7}{2}}\times 8)$$

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

$$\frac{7}{2}\times 8x^{\frac{7}{2}-1}$$

Multiply \frac{7}{2} times 8.

$$28x^{\frac{7}{2}-1}$$

Subtract 1 from \frac{7}{2}.

$$28x^{\frac{5}{2}}$$