Solve (4b)^1/2(8b^1/4)

$$( 4 b ) ^ { 1 / 2 } ( 8 b ^ { 1 / 4 } )$$

Evaluate

$$16b^{\frac{3}{4}}$$

Solution Steps

Expand \left(4b\right)^{\frac{1}{2}}.

$$4^{\frac{1}{2}}b^{\frac{1}{2}}\times 8b^{\frac{1}{4}}$$

Calculate 4 to the power of \frac{1}{2} and get 2.

$$2b^{\frac{1}{2}}\times 8b^{\frac{1}{4}}$$

Multiply 2 and 8 to get 16.

$$16b^{\frac{1}{2}}b^{\frac{1}{4}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{4} to get \frac{3}{4}.

$$16b^{\frac{3}{4}}$$

Differentiate w.r.t. b

$$\frac{12}{\sqrt[4]{b}}$$

Steps Using Definition of a Derivative

Expand \left(4b\right)^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(4^{\frac{1}{2}}b^{\frac{1}{2}}\times 8b^{\frac{1}{4}})$$

Calculate 4 to the power of \frac{1}{2} and get 2.

$$\frac{\mathrm{d}}{\mathrm{d}b}(2b^{\frac{1}{2}}\times 8b^{\frac{1}{4}})$$

Multiply 2 and 8 to get 16.

$$\frac{\mathrm{d}}{\mathrm{d}b}(16b^{\frac{1}{2}}b^{\frac{1}{4}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{4} to get \frac{3}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}b}(16b^{\frac{3}{4}})$$

The derivative of ax^{n} is nax^{n-1}.

$$\frac{3}{4}\times 16b^{\frac{3}{4}-1}$$

Multiply \frac{3}{4} times 16.

$$12b^{\frac{3}{4}-1}$$

Subtract 1 from \frac{3}{4}.

$$12b^{-\frac{1}{4}}$$