Solve 2x^frac{1{2}}(2x^2)^2

$$2 x ^ { \frac { 1 } { 2 } } ( 2 x ^ { 2 } ) ^ { 2 }$$

Evaluate

$$8x^{\frac{9}{2}}$$

Solution Steps

Expand \left(2x^{2}\right)^{2}.

$$2x^{\frac{1}{2}}\times 2^{2}\left(x^{2}\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$2x^{\frac{1}{2}}\times 2^{2}x^{4}$$

Calculate 2 to the power of 2 and get 4.

$$2x^{\frac{1}{2}}\times 4x^{4}$$

Multiply 2 and 4 to get 8.

$$8x^{\frac{1}{2}}x^{4}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and 4 to get \frac{9}{2}.

$$8x^{\frac{9}{2}}$$

Differentiate w.r.t. x

$$36x^{\frac{7}{2}}$$

Steps Using Definition of a Derivative

Expand \left(2x^{2}\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{\frac{1}{2}}\times 2^{2}\left(x^{2}\right)^{2})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{\frac{1}{2}}\times 2^{2}x^{4})$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(2x^{\frac{1}{2}}\times 4x^{4})$$

Multiply 2 and 4 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(8x^{\frac{1}{2}}x^{4})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and 4 to get \frac{9}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(8x^{\frac{9}{2}})$$

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

$$\frac{9}{2}\times 8x^{\frac{9}{2}-1}$$

Multiply \frac{9}{2} times 8.

$$36x^{\frac{9}{2}-1}$$

Subtract 1 from \frac{9}{2}.

$$36x^{\frac{7}{2}}$$