Solve frac{b^-frac{-1{2}}b^frac{-1{2}}}{b^frac{1{3}}}

$$\frac { b ^ { - \frac { - 1 } { 2 } } b ^ { \frac { - 1 } { 2 } } } { b ^ { \frac { 1 } { 3 } } }$$

Evaluate

$$\frac{1}{\sqrt[3]{b}}$$

Steps for Raising Power to Another Power

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$\frac{b^{-\left(-\frac{1}{2}\right)}b^{\frac{-1}{2}}}{b^{\frac{1}{3}}}$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$\frac{b^{\frac{1}{2}}b^{\frac{-1}{2}}}{b^{\frac{1}{3}}}$$

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$\frac{b^{\frac{1}{2}}b^{-\frac{1}{2}}}{b^{\frac{1}{3}}}$$

Multiply b^{\frac{1}{2}} and b^{-\frac{1}{2}} to get 1.

$$\frac{1}{b^{\frac{1}{3}}}$$

Use the rules of exponents to simplify the expression.

$$\frac{1}{\sqrt[3]{b}}$$

Differentiate w.r.t. b

$$-\frac{1}{3b^{\frac{4}{3}}}$$

Steps Using Chain Rule

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{-\left(-\frac{1}{2}\right)}b^{\frac{-1}{2}}}{b^{\frac{1}{3}}})$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{\frac{1}{2}}b^{\frac{-1}{2}}}{b^{\frac{1}{3}}})$$

Fraction \frac{-1}{2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{b^{\frac{1}{2}}b^{-\frac{1}{2}}}{b^{\frac{1}{3}}})$$

Multiply b^{\frac{1}{2}} and b^{-\frac{1}{2}} to get 1.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{1}{b^{\frac{1}{3}}})$$

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).

$$-\left(\sqrt[3]{b}\right)^{-1-1}\frac{\mathrm{d}}{\mathrm{d}b}(\sqrt[3]{b})$$

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}.

$$-\left(\sqrt[3]{b}\right)^{-2}\times \frac{1}{3}b^{\frac{1}{3}-1}$$

Simplify.

$$-\frac{1}{3}b^{-\frac{2}{3}}\left(\sqrt[3]{b}\right)^{-2}$$