Solve (frac{x^frac{1{2}}x^-frac{5{4}}}{x^frac{3{4}}})^frac{4{3}}

$$( \frac { x ^ { \frac { 1 } { 2 } } x ^ { - \frac { 5 } { 4 } } } { x ^ { \frac { 3 } { 4 } } } ) ^ { \frac { 4 } { 3 } }$$

Evaluate

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

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and -\frac{5}{4} to get -\frac{3}{4}.

$$\left(\frac{x^{-\frac{3}{4}}}{x^{\frac{3}{4}}}\right)^{\frac{4}{3}}$$

Rewrite x^{\frac{3}{4}} as x^{-\frac{3}{4}}x^{\frac{3}{2}}. Cancel out x^{-\frac{3}{4}} in both numerator and denominator.

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

To raise \frac{1}{x^{\frac{3}{2}}} to a power, raise both numerator and denominator to the power and then divide.

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

To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and \frac{4}{3} to get 2.

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

Calculate 1 to the power of \frac{4}{3} and get 1.

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and -\frac{5}{4} to get -\frac{3}{4}.

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

Rewrite x^{\frac{3}{4}} as x^{-\frac{3}{4}}x^{\frac{3}{2}}. Cancel out x^{-\frac{3}{4}} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\frac{1}{x^{\frac{3}{2}}}\right)^{\frac{4}{3}})$$

To raise \frac{1}{x^{\frac{3}{2}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{\frac{4}{3}}}{\left(x^{\frac{3}{2}}\right)^{\frac{4}{3}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{2} and \frac{4}{3} to get 2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1^{\frac{4}{3}}}{x^{2}})$$

Calculate 1 to the power of \frac{4}{3} and get 1.

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

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

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

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

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

Simplify.

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

For any term t, t^{1}=t.

$$-2x\left(x^{2}\right)^{-2}$$