Solve (9^4b^-5/b^2)^-1

$$( \frac { 9 ^ { 4 } b ^ { - 5 } } { b ^ { 2 } } ) ^ { - 1 }$$

Evaluate

$$\frac{b^{7}}{6561}$$

Solution Steps

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

$$\left(\frac{9^{4}}{b^{7}}\right)^{-1}$$

Calculate 9 to the power of 4 and get 6561.

$$\left(\frac{6561}{b^{7}}\right)^{-1}$$

To raise \frac{6561}{b^{7}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{6561^{-1}}{\left(b^{7}\right)^{-1}}$$

To raise a power to another power, multiply the exponents. Multiply 7 and -1 to get -7.

$$\frac{6561^{-1}}{b^{-7}}$$

Calculate 6561 to the power of -1 and get \frac{1}{6561}.

$$\frac{\frac{1}{6561}}{b^{-7}}$$

Express \frac{\frac{1}{6561}}{b^{-7}} as a single fraction.

$$\frac{1}{6561b^{-7}}$$

Differentiate w.r.t. b

$$\frac{7b^{6}}{6561}$$

Steps Using Chain Rule

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\left(\frac{9^{4}}{b^{7}}\right)^{-1})$$

Calculate 9 to the power of 4 and get 6561.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\left(\frac{6561}{b^{7}}\right)^{-1})$$

To raise \frac{6561}{b^{7}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{6561^{-1}}{\left(b^{7}\right)^{-1}})$$

To raise a power to another power, multiply the exponents. Multiply 7 and -1 to get -7.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{6561^{-1}}{b^{-7}})$$

Calculate 6561 to the power of -1 and get \frac{1}{6561}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{\frac{1}{6561}}{b^{-7}})$$

Express \frac{\frac{1}{6561}}{b^{-7}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{6561b^{-7}})$$

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).

$$-\left(6561b^{-7}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}b}(6561b^{-7})$$

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}.

$$-\left(6561b^{-7}\right)^{-2}\left(-7\right)\times 6561b^{-7-1}$$

Simplify.

$$45927b^{-8}\times \left(6561b^{-7}\right)^{-2}$$