Solve g^-1*g^8/g^-57*g^81

$$\frac { g ^ { - 1 } \cdot g ^ { 8 } } { g ^ { - 57 } \cdot g ^ { 81 } }$$

Evaluate

$$\frac{1}{g^{17}}$$

Solution Steps

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

$$\frac{g^{7}}{g^{-57}g^{81}}$$

To multiply powers of the same base, add their exponents. Add -57 and 81 to get 24.

$$\frac{g^{7}}{g^{24}}$$

Rewrite g^{24} as g^{7}g^{17}. Cancel out g^{7} in both numerator and denominator.

$$\frac{1}{g^{17}}$$

Differentiate w.r.t. g

$$-\frac{17}{g^{18}}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}g}(\frac{g^{7}}{g^{-57}g^{81}})$$

To multiply powers of the same base, add their exponents. Add -57 and 81 to get 24.

$$\frac{\mathrm{d}}{\mathrm{d}g}(\frac{g^{7}}{g^{24}})$$

Rewrite g^{24} as g^{7}g^{17}. Cancel out g^{7} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}g}(\frac{1}{g^{17}})$$

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).

$$-\left(g^{17}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}g}(g^{17})$$

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}.

$$-\left(g^{17}\right)^{-2}\times 17g^{17-1}$$

Simplify.

$$-17g^{16}\left(g^{17}\right)^{-2}$$