Solve 3sqrt{5}(sqrt{5}+x)-2x(x^2-3sqrt{5})

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

Evaluate

$$-2x^{3}+9\sqrt{5}x+15$$

Solution Steps

Use the distributive property to multiply 3\sqrt{5} by \sqrt{5}+x.

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

The square of \sqrt{5} is 5.

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

Multiply 3 and 5 to get 15.

$$15+3\sqrt{5}x-2x\left(x^{2}-3\sqrt{5}\right)$$

Use the distributive property to multiply -2x by x^{2}-3\sqrt{5}.

$$15+3\sqrt{5}x-2x^{3}+6x\sqrt{5}$$

Combine 3\sqrt{5}x and 6x\sqrt{5} to get 9\sqrt{5}x.

$$15+9\sqrt{5}x-2x^{3}$$

Differentiate w.r.t. x

$$-6x^{2}+9\sqrt{5}$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply 3\sqrt{5} by \sqrt{5}+x.

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

The square of \sqrt{5} is 5.

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

Multiply 3 and 5 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(15+3\sqrt{5}x-2x\left(x^{2}-3\sqrt{5}\right))$$

Use the distributive property to multiply -2x by x^{2}-3\sqrt{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(15+3\sqrt{5}x-2x^{3}+6x\sqrt{5})$$

Combine 3\sqrt{5}x and 6x\sqrt{5} to get 9\sqrt{5}x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(15+9\sqrt{5}x-2x^{3})$$

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}.

$$9\sqrt{5}x^{1-1}+3\left(-2\right)x^{3-1}$$

Subtract 1 from 1.

$$9\sqrt{5}x^{0}+3\left(-2\right)x^{3-1}$$

Multiply 3 times -2.

$$9\sqrt{5}x^{0}-6x^{3-1}$$

Subtract 1 from 3.

$$9\sqrt{5}x^{0}-6x^{2}$$

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

$$9\sqrt{5}\times 1-6x^{2}$$

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

$$9\sqrt{5}-6x^{2}$$