Solve (2b^4)^-1

$$( 2 b ^ { 4 } ) ^ { - 1 }$$

Evaluate

$$\frac{1}{2b^{4}}$$

Steps for Raising Power to Another Power

Use the rules of exponents to simplify the expression.

$$\frac{1}{2b^{4}}$$

To raise the product of two or more numbers to a power, raise each number to the power and take their product.

$$\frac{1}{2}\times \frac{1}{b^{4}}$$

Differentiate w.r.t. b

$$-\frac{2}{b^{5}}$$

Steps Using Chain Rule

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).

$$-\left(2b^{4}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}b}(2b^{4})$$

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}.

$$-\left(2b^{4}\right)^{-2}\times 4\times 2b^{4-1}$$

Simplify.

$$-8b^{3}\times \left(2b^{4}\right)^{-2}$$