Solve frac{t^-frac{3{2}}*t^-frac{5{2}}}{t^-frac{3{2}}}

$$\frac { t ^ { - \frac { 3 } { 2 } } \cdot t ^ { - \frac { 5 } { 2 } } } { t ^ { - \frac { 3 } { 2 } } }$$

Differentiate w.r.t. t

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

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add -\frac{3}{2} and -\frac{5}{2} to get -4.

$$\frac{\mathrm{d}}{\mathrm{d}t}(\frac{t^{-4}}{t^{-\frac{3}{2}}})$$

Rewrite t^{-\frac{3}{2}} as t^{-4}t^{\frac{5}{2}}. Cancel out t^{-4} in both numerator and denominator.

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

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).

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

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}.

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

Simplify.

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

Evaluate

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