Solve frac{{left(4x^0.5right)}^frac{2{2}}}{{x}^-frac{1{4}}}

$$\frac{ { \left(4 { x }^{ 0.5 } \right) }^{ \frac{ 2 }{ 2 } } }{ { x }^{ - \frac{ 1 }{ 4 } } }$$

Evaluate

$$4x^{0.75}$$

Solution Steps

Divide 2 by 2 to get 1.

$$\frac{\left(4x^{0.5}\right)^{1}}{x^{-\frac{1}{4}}}$$

Calculate 4x^{0.5} to the power of 1 and get 4x^{0.5}.

$$\frac{4x^{0.5}}{x^{-\frac{1}{4}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

Differentiate w.r.t. x

$$\frac{3}{\sqrt[4]{x}}$$

Steps Using Definition of a Derivative

Divide 2 by 2 to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(4x^{0.5}\right)^{1}}{x^{-\frac{1}{4}}})$$

Calculate 4x^{0.5} to the power of 1 and get 4x^{0.5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{0.5}}{x^{-\frac{1}{4}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

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

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

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

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

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

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

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