Solve (1/a^frac{1{2}})^5

$$( \frac { 1 } { a ^ { \frac { 1 } { 2 } } } ) ^ { 5 }$$

Differentiate w.r.t. a

$$-\frac{5}{2a^{\frac{7}{2}}}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{5}}{\left(a^{\frac{1}{2}}\right)^{5}})$$

To raise a power to another power, multiply the exponents. Multiply \frac{1}{2} and 5 to get \frac{5}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1^{5}}{a^{\frac{5}{2}}})$$

Calculate 1 to the power of 5 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{a^{\frac{5}{2}}})$$

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).

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

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}.

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

Simplify.

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

Evaluate

$$\frac{1}{a^{\frac{5}{2}}}$$