Solve (1/a^frac{5{6}})^frac{4{3}}

$$( \frac { 1 } { a ^ { \frac { 5 } { 6 } } } ) ^ { \frac { 4 } { 3 } }$$

Differentiate w.r.t. a

$$-\frac{10}{9a^{\frac{19}{9}}}$$

Steps Using Chain Rule

To raise \frac{1}{a^{\frac{5}{6}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{\frac{4}{3}}}{\left(a^{\frac{5}{6}}\right)^{\frac{4}{3}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{5}{6} and \frac{4}{3} to get \frac{10}{9}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{\frac{4}{3}}}{a^{\frac{10}{9}}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{\frac{10}{9}}})$$

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).

$$-\left(a^{\frac{10}{9}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}a}(a^{\frac{10}{9}})$$

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}.

$$-\left(a^{\frac{10}{9}}\right)^{-2}\times \frac{10}{9}a^{\frac{10}{9}-1}$$

Simplify.

$$-\frac{10}{9}\sqrt[9]{a}\left(a^{\frac{10}{9}}\right)^{-2}$$

Evaluate

$$\frac{1}{a^{\frac{10}{9}}}$$