Solve (1/a^frac{7{3}})^-frac{5{7}}

$$( \frac { 1 } { a ^ { \frac { 7 } { 3 } } } ) ^ { - \frac { 5 } { 7 } }$$

Evaluate

$$a^{\frac{5}{3}}$$

Steps for Raising Quotient to a Power

Use the rules of exponents to simplify the expression.

$$\left(\frac{1}{a^{\frac{7}{3}}}\right)^{-\frac{5}{7}}$$

To raise the quotient of two numbers to a power, raise each number to the power and then divide.

$$\frac{1^{-\frac{5}{7}}}{\left(a^{\frac{7}{3}}\right)^{-\frac{5}{7}}}$$

To raise a power to another power, multiply the exponents.

$$\frac{1}{a^{\frac{7}{3}\left(-\frac{5}{7}\right)}}$$

Multiply \frac{7}{3} times -\frac{5}{7} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$\frac{a^{\frac{5}{3}}}{1}$$

Differentiate w.r.t. a

$$\frac{5a^{\frac{2}{3}}}{3}$$

Steps Using Chain Rule

To raise \frac{1}{a^{\frac{7}{3}}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{-\frac{5}{7}}}{\left(a^{\frac{7}{3}}\right)^{-\frac{5}{7}}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{7}{3} and -\frac{5}{7} to get -\frac{5}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{-\frac{5}{7}}}{a^{-\frac{5}{3}}})$$

Calculate 1 to the power of -\frac{5}{7} and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{-\frac{5}{3}}})$$

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).

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

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}.

$$-\left(a^{-\frac{5}{3}}\right)^{-2}\left(-\frac{5}{3}\right)a^{-\frac{5}{3}-1}$$

Simplify.

$$\frac{5}{3}a^{-\frac{8}{3}}\left(a^{-\frac{5}{3}}\right)^{-2}$$