Solve r^-5*r^-1/r^8*r^-5

$$\frac { r ^ { - 5 } \cdot r ^ { - 1 } } { r ^ { 8 } \cdot r ^ { - 5 } }$$

Evaluate

$$\frac{1}{r^{9}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -5 and -1 to get -6.

$$\frac{r^{-6}}{r^{8}r^{-5}}$$

To multiply powers of the same base, add their exponents. Add 8 and -5 to get 3.

$$\frac{r^{-6}}{r^{3}}$$

Rewrite r^{3} as r^{-6}r^{9}. Cancel out r^{-6} in both numerator and denominator.

$$\frac{1}{r^{9}}$$

Differentiate w.r.t. r

$$-\frac{9}{r^{10}}$$

Steps Using Chain Rule

To multiply powers of the same base, add their exponents. Add -5 and -1 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{r^{-6}}{r^{8}r^{-5}})$$

To multiply powers of the same base, add their exponents. Add 8 and -5 to get 3.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{r^{-6}}{r^{3}})$$

Rewrite r^{3} as r^{-6}r^{9}. Cancel out r^{-6} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}r}(\frac{1}{r^{9}})$$

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).

$$-\left(r^{9}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}r}(r^{9})$$

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}.

$$-\left(r^{9}\right)^{-2}\times 9r^{9-1}$$

Simplify.

$$-9r^{8}\left(r^{9}\right)^{-2}$$