Solve (3{x}^frac{1{2}})(-2{x}^frac{5{2}})

$$(3 { x }^{ \frac{ 1 }{ 2 } } )(-2 { x }^{ \frac{ 5 }{ 2 } } )$$

Evaluate

$$-6x^{3}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{5}{2} to get 3.

$$3x^{3}\left(-2\right)$$

Multiply 3 and -2 to get -6.

$$-6x^{3}$$

Differentiate w.r.t. x

$$-18x^{2}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{5}{2} to get 3.

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

Multiply 3 and -2 to get -6.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-6x^{3})$$

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

$$3\left(-6\right)x^{3-1}$$

Multiply 3 times -6.

$$-18x^{3-1}$$

Subtract 1 from 3.

$$-18x^{2}$$