Solve 2x^3(25x^2/4y^4)^frac{1{2}}

$$2 x ^ { 3 } ( \frac { 25 x ^ { 2 } } { 4 y ^ { 4 } } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$\frac{5|x|x^{3}}{y^{2}}$$

Solution Steps

To raise \frac{25x^{2}}{4y^{4}} to a power, raise both numerator and denominator to the power and then divide.

$$2x^{3}\times \frac{\left(25x^{2}\right)^{\frac{1}{2}}}{\left(4y^{4}\right)^{\frac{1}{2}}}$$

Expand \left(25x^{2}\right)^{\frac{1}{2}}.

$$2x^{3}\times \frac{25^{\frac{1}{2}}\left(x^{2}\right)^{\frac{1}{2}}}{\left(4y^{4}\right)^{\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{1}{2} to get 1.

$$2x^{3}\times \frac{25^{\frac{1}{2}}x^{1}}{\left(4y^{4}\right)^{\frac{1}{2}}}$$

Calculate 25 to the power of \frac{1}{2} and get 5.

$$2x^{3}\times \frac{5x^{1}}{\left(4y^{4}\right)^{\frac{1}{2}}}$$

Calculate x to the power of 1 and get x.

$$2x^{3}\times \frac{5x}{\left(4y^{4}\right)^{\frac{1}{2}}}$$

Expand \left(4y^{4}\right)^{\frac{1}{2}}.

$$2x^{3}\times \frac{5x}{4^{\frac{1}{2}}\left(y^{4}\right)^{\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and \frac{1}{2} to get 2.

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

Calculate 4 to the power of \frac{1}{2} and get 2.

$$2x^{3}\times \frac{5x}{2y^{2}}$$

Express 2\times \frac{5x}{2y^{2}} as a single fraction.

$$\frac{2\times 5x}{2y^{2}}x^{3}$$

Cancel out 2 in both numerator and denominator.

$$\frac{5x}{y^{2}}x^{3}$$

Express \frac{5x}{y^{2}}x^{3} as a single fraction.

$$\frac{5xx^{3}}{y^{2}}$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{5x^{4}}{y^{2}}$$

Differentiate w.r.t. x

$$20\times \left(\frac{x}{y}\right)^{2}|x|$$