Solve (1/h^-1*h^-7)^frac{3{20}}

$$( \frac { 1 } { h ^ { - 1 } \cdot h ^ { - 7 } } ) ^ { \frac { 3 } { 20 } }$$

Differentiate w.r.t. h

$$\frac{6\sqrt[5]{h}}{5}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add -1 and -7 to get -8.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\left(\frac{1}{h^{-8}}\right)^{\frac{3}{20}})$$

To raise \frac{1}{h^{-8}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{1^{\frac{3}{20}}}{\left(h^{-8}\right)^{\frac{3}{20}}})$$

To raise a power to another power, multiply the exponents. Multiply -8 and \frac{3}{20} to get -\frac{6}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{1^{\frac{3}{20}}}{h^{-\frac{6}{5}}})$$

Calculate 1 to the power of \frac{3}{20} and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}h}(\frac{1}{h^{-\frac{6}{5}}})$$

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).

$$-\left(h^{-\frac{6}{5}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}h}(h^{-\frac{6}{5}})$$

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}.

$$-\left(h^{-\frac{6}{5}}\right)^{-2}\left(-\frac{6}{5}\right)h^{-\frac{6}{5}-1}$$

Simplify.

$$\frac{6}{5}h^{-\frac{11}{5}}\left(h^{-\frac{6}{5}}\right)^{-2}$$

Evaluate

$$h^{\frac{6}{5}}$$