Solve (sqrt{3}x+sqrt{5})(sqrt{3}x-sqrt{5})

$$( \sqrt { 3 } x + \sqrt { 5 } ) ( \sqrt { 3 } x - \sqrt { 5 } )$$

Evaluate

$$3x^{2}-5$$

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}.

$$\left(\sqrt{3}x\right)^{2}-\left(\sqrt{5}\right)^{2}$$

Expand \left(\sqrt{3}x\right)^{2}.

$$\left(\sqrt{3}\right)^{2}x^{2}-\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{3} is 3.

$$3x^{2}-\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$3x^{2}-5$$

Differentiate w.r.t. x

$$6x$$

Steps Using Derivative Rule for Sum

Consider \left(\sqrt{3}x+\sqrt{5}\right)\left(\sqrt{3}x-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\sqrt{3}x\right)^{2}-\left(\sqrt{5}\right)^{2})$$

Expand \left(\sqrt{3}x\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(\sqrt{3}\right)^{2}x^{2}-\left(\sqrt{5}\right)^{2})$$

The square of \sqrt{3} is 3.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}-\left(\sqrt{5}\right)^{2})$$

The square of \sqrt{5} is 5.

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

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}.

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

Multiply 2 times 3.

$$6x^{2-1}$$

Subtract 1 from 2.

$$6x^{1}$$

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

$$6x$$