Solve 4xsqrt{x}1/2sqrt{x}(2x^2+1)

$$4x \sqrt{ x } \frac{ 1 }{ 2 \sqrt{ x } } (2 { x }^{ 2 } +1)$$

Evaluate

$$4x^{3}+2x$$

Solution Steps

Express 4\times \frac{1}{2\sqrt{x}} as a single fraction.

$$\frac{4}{2\sqrt{x}}x\sqrt{x}\left(2x^{2}+1\right)$$

Express \frac{4}{2\sqrt{x}}\left(2x^{2}+1\right) as a single fraction.

$$\frac{4\left(2x^{2}+1\right)}{2\sqrt{x}}x\sqrt{x}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\left(2x^{2}+1\right)}{\sqrt{x}}x\sqrt{x}$$

Use the distributive property to multiply 2 by 2x^{2}+1.

$$\frac{4x^{2}+2}{\sqrt{x}}x\sqrt{x}$$

Express \frac{4x^{2}+2}{\sqrt{x}}x as a single fraction.

$$\frac{\left(4x^{2}+2\right)x}{\sqrt{x}}\sqrt{x}$$

Cancel out \sqrt{x} and \sqrt{x}.

$$\left(4x^{2}+2\right)x$$

Use the distributive property to multiply 4x^{2}+2 by x.

$$4x^{3}+2x$$

Differentiate w.r.t. x

$$12x^{2}+2$$

Steps Using Derivative Rule for Sum

Express 4\times \frac{1}{2\sqrt{x}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4}{2\sqrt{x}}x\sqrt{x}\left(2x^{2}+1\right))$$

Express \frac{4}{2\sqrt{x}}\left(2x^{2}+1\right) as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4\left(2x^{2}+1\right)}{2\sqrt{x}}x\sqrt{x})$$

Cancel out 2 in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2\left(2x^{2}+1\right)}{\sqrt{x}}x\sqrt{x})$$

Use the distributive property to multiply 2 by 2x^{2}+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x^{2}+2}{\sqrt{x}}x\sqrt{x})$$

Express \frac{4x^{2}+2}{\sqrt{x}}x as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\left(4x^{2}+2\right)x}{\sqrt{x}}\sqrt{x})$$

Cancel out \sqrt{x} and \sqrt{x}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(4x^{2}+2\right)x)$$

Use the distributive property to multiply 4x^{2}+2 by x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{3}+2x)$$

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}.

$$3\times 4x^{3-1}+2x^{1-1}$$

Multiply 3 times 4.

$$12x^{3-1}+2x^{1-1}$$

Subtract 1 from 3.

$$12x^{2}+2x^{1-1}$$

Subtract 1 from 1.

$$12x^{2}+2x^{0}$$

For any term t except 0, t^{0}=1.

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

For any term t, t\times 1=t and 1t=t.

$$12x^{2}+2$$