Solve 1/2:b^frac{1{2}}

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

Differentiate w.r.t. b

$$-\frac{1}{4b^{\frac{3}{2}}}$$

Steps Using Chain Rule

Express \frac{\frac{1}{2}}{b^{\frac{1}{2}}} as a single fraction.

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

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).

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

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}.

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

Simplify.

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

Evaluate

$$\frac{1}{2\sqrt{b}}$$

Solution Steps

Express \frac{\frac{1}{2}}{b^{\frac{1}{2}}} as a single fraction.

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