Solve r^3s^2/r^3s^4

$$\frac { r ^ { 3 } s ^ { 2 } } { r ^ { 3 } s ^ { 4 } }$$

Evaluate

$$\frac{1}{s^{2}}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{r^{3}s^{2}}{r^{3}s^{4}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$r^{3-3}s^{2-4}$$

Subtract 3 from 3.

$$r^{0}s^{2-4}$$

Subtract 4 from 2.

$$r^{0}s^{-2}$$

For any number a except 0, a^{0}=1.

$$\frac{1}{s^{2}}$$

Differentiate w.r.t. s

$$-\frac{2}{s^{3}}$$

Steps Using Chain Rule

Cancel out s^{2}r^{3} in both numerator and denominator.

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

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).

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

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}.

$$-\left(s^{2}\right)^{-2}\times 2s^{2-1}$$

Simplify.

$$-2s^{1}\left(s^{2}\right)^{-2}$$

For any term t, t^{1}=t.

$$-2s\left(s^{2}\right)^{-2}$$