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

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

Differentiate w.r.t. a

$$-\frac{3}{4a^{\frac{7}{4}}}$$

Steps Using Chain Rule

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

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

Multiply a^{-\frac{1}{2}} and a^{\frac{1}{2}} to get 1.

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

Calculate 1 to the power of -\frac{2}{3} and get 1.

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

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).

$$-\left(a^{\frac{3}{4}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{3}{4}})$$

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}.

$$-\left(a^{\frac{3}{4}}\right)^{-2}\times \frac{3}{4}a^{\frac{3}{4}-1}$$

Simplify.

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

Evaluate

$$\frac{1}{a^{\frac{3}{4}}}$$