Solve (st/st^4)^frac{9{5}}

$$( \frac { s t } { s t ^ { 4 } } ) ^ { \frac { 9 } { 5 } }$$

Evaluate

$$\frac{1}{t^{\frac{27}{5}}}$$

Steps for Raising Power to Another Power

Cancel out st in both numerator and denominator.

$$\left(\frac{1}{t^{3}}\right)^{\frac{9}{5}}$$

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

$$\frac{1^{\frac{9}{5}}}{\left(t^{3}\right)^{\frac{9}{5}}}$$

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{9}{5} to get \frac{27}{5}.

$$\frac{1^{\frac{9}{5}}}{t^{\frac{27}{5}}}$$

Calculate 1 to the power of \frac{9}{5} and get 1.

$$\frac{1}{t^{\frac{27}{5}}}$$

Differentiate w.r.t. t

$$-\frac{27}{5t^{\frac{32}{5}}}$$

Steps Using Chain Rule

Cancel out st in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\left(\frac{1}{t^{3}}\right)^{\frac{9}{5}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1^{\frac{9}{5}}}{\left(t^{3}\right)^{\frac{9}{5}}})$$

To raise a power to another power, multiply the exponents. Multiply 3 and \frac{9}{5} to get \frac{27}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1^{\frac{9}{5}}}{t^{\frac{27}{5}}})$$

Calculate 1 to the power of \frac{9}{5} and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{1}{t^{\frac{27}{5}}})$$

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).

$$-\left(t^{\frac{27}{5}}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}t}(t^{\frac{27}{5}})$$

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}.

$$-\left(t^{\frac{27}{5}}\right)^{-2}\times \frac{27}{5}t^{\frac{27}{5}-1}$$

Simplify.

$$-\frac{27}{5}t^{\frac{22}{5}}\left(t^{\frac{27}{5}}\right)^{-2}$$