Solve (2sqrt{x}+sqrt{3})(2sqrt{x}-sqrt{3})

$$( 2 \sqrt { x } + \sqrt { 3 } ) ( 2 \sqrt { x } - \sqrt { 3 } )$$

Evaluate

$$4x-3$$

Solution Steps

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(2\sqrt{x}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x}\right)^{2}-\left(\sqrt{3}\right)^{2}$$

Calculate \sqrt{x} to the power of 2 and get x.

$$4x-\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$4x-3$$

Differentiate w.r.t. x

$$4$$

Steps Using Derivative Rule for Sum

Consider \left(2\sqrt{x}+\sqrt{3}\right)\left(2\sqrt{x}-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(2\sqrt{x}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4\left(\sqrt{x}\right)^{2}-\left(\sqrt{3}\right)^{2})$$

Calculate \sqrt{x} to the power of 2 and get x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x-\left(\sqrt{3}\right)^{2})$$

The square of \sqrt{3} is 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x-3)$$

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}.

$$4x^{1-1}$$

Subtract 1 from 1.

$$4x^{0}$$

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

$$4\times 1$$

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

$$4$$