Solve (36x)^1/2(2x^3/2)

$$( 36 x ) ^ { 1 / 2 } ( 2 x ^ { 3 / 2 } )$$

Evaluate

$$12x^{2}$$

Solution Steps

Expand \left(36x\right)^{\frac{1}{2}}.

$$36^{\frac{1}{2}}x^{\frac{1}{2}}\times 2x^{\frac{3}{2}}$$

Calculate 36 to the power of \frac{1}{2} and get 6.

$$6x^{\frac{1}{2}}\times 2x^{\frac{3}{2}}$$

Multiply 6 and 2 to get 12.

$$12x^{\frac{1}{2}}x^{\frac{3}{2}}$$

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

$$12x^{2}$$

Differentiate w.r.t. x

$$24x$$

Steps Using Definition of a Derivative

Expand \left(36x\right)^{\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(36^{\frac{1}{2}}x^{\frac{1}{2}}\times 2x^{\frac{3}{2}})$$

Calculate 36 to the power of \frac{1}{2} and get 6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(6x^{\frac{1}{2}}\times 2x^{\frac{3}{2}})$$

Multiply 6 and 2 to get 12.

$$\frac{\mathrm{d}}{\mathrm{d}x}(12x^{\frac{1}{2}}x^{\frac{3}{2}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(12x^{2})$$

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

$$2\times 12x^{2-1}$$

Multiply 2 times 12.

$$24x^{2-1}$$

Subtract 1 from 2.

$$24x^{1}$$

For any term t, t^{1}=t.

$$24x$$