Solve 4(2-x)^2*sqrt{3}x

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

Evaluate

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

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-x\right)^{2}.

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

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

$$\left(16-16x+4x^{2}\right)\sqrt{3}x$$

Use the distributive property to multiply 16-16x+4x^{2} by \sqrt{3}.

$$\left(16\sqrt{3}-16x\sqrt{3}+4x^{2}\sqrt{3}\right)x$$

Use the distributive property to multiply 16\sqrt{3}-16x\sqrt{3}+4x^{2}\sqrt{3} by x.

$$16\sqrt{3}x-16\sqrt{3}x^{2}+4\sqrt{3}x^{3}$$

Differentiate w.r.t. x

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

Steps Using Derivative Rule for Sum

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2-x\right)^{2}.

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(16-16x+4x^{2}\right)\sqrt{3}x)$$

Use the distributive property to multiply 16-16x+4x^{2} by \sqrt{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(16\sqrt{3}-16x\sqrt{3}+4x^{2}\sqrt{3}\right)x)$$

Use the distributive property to multiply 16\sqrt{3}-16x\sqrt{3}+4x^{2}\sqrt{3} by x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16\sqrt{3}x-16\sqrt{3}x^{2}+4\sqrt{3}x^{3})$$

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}.

$$16\sqrt{3}x^{1-1}+2\left(-16\sqrt{3}\right)x^{2-1}+3\times 4\sqrt{3}x^{3-1}$$

Subtract 1 from 1.

$$16\sqrt{3}x^{0}+2\left(-16\sqrt{3}\right)x^{2-1}+3\times 4\sqrt{3}x^{3-1}$$

Multiply 2 times -16\sqrt{3}.

$$16\sqrt{3}x^{0}+\left(-32\sqrt{3}\right)x^{2-1}+3\times 4\sqrt{3}x^{3-1}$$

Subtract 1 from 2.

$$16\sqrt{3}x^{0}+\left(-32\sqrt{3}\right)x^{1}+3\times 4\sqrt{3}x^{3-1}$$

Multiply 2 times -16\sqrt{3}.

$$16\sqrt{3}x^{0}+\left(-32\sqrt{3}\right)x^{1}+12\sqrt{3}x^{3-1}$$

Subtract 1 from 3.

$$16\sqrt{3}x^{0}+\left(-32\sqrt{3}\right)x^{1}+12\sqrt{3}x^{2}$$

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

$$16\sqrt{3}x^{0}+\left(-32\sqrt{3}\right)x+12\sqrt{3}x^{2}$$

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

$$16\sqrt{3}\times 1+\left(-32\sqrt{3}\right)x+12\sqrt{3}x^{2}$$

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

$$16\sqrt{3}+\left(-32\sqrt{3}\right)x+12\sqrt{3}x^{2}$$