Solve 5x^3*2{x}^frac{1{2}}

$$5 { x }^{ 3 } \times 2 { x }^{ \frac{ 1 }{ 2 } }$$

Evaluate

$$10x^{\frac{7}{2}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and \frac{1}{2} to get \frac{7}{2}.

$$5x^{\frac{7}{2}}\times 2$$

Multiply 5 and 2 to get 10.

$$10x^{\frac{7}{2}}$$

Differentiate w.r.t. x

$$35x^{\frac{5}{2}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and \frac{1}{2} to get \frac{7}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{\frac{7}{2}}\times 2)$$

Multiply 5 and 2 to get 10.

$$\frac{\mathrm{d}}{\mathrm{d}x}(10x^{\frac{7}{2}})$$

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

$$\frac{7}{2}\times 10x^{\frac{7}{2}-1}$$

Multiply \frac{7}{2} times 10.

$$35x^{\frac{7}{2}-1}$$

Subtract 1 from \frac{7}{2}.

$$35x^{\frac{5}{2}}$$