Solve d/dxleft(3left(3x+x+1right)^2(2x+1)right)

$$\frac{d}{d x } \left(3 { \left(3x+x+1 \right) }^{ 2 } (2x+1) \right)$$

Evaluate

$$288x^{2}+192x+30$$

Steps Using Derivative Rule for Sum

Combine 3x and x to get 4x.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3\left(4x+1\right)^{2}\left(2x+1\right))$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+1\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(3\left(16x^{2}+8x+1\right)\left(2x+1\right))$$

Use the distributive property to multiply 3 by 16x^{2}+8x+1.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(48x^{2}+24x+3\right)\left(2x+1\right))$$

Use the distributive property to multiply 48x^{2}+24x+3 by 2x+1 and combine like terms.

$$\frac{\mathrm{d}}{\mathrm{d}x}(96x^{3}+96x^{2}+30x+3)$$

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}.

$$3\times 96x^{3-1}+2\times 96x^{2-1}+30x^{1-1}$$

Multiply 3 times 96.

$$288x^{3-1}+2\times 96x^{2-1}+30x^{1-1}$$

Subtract 1 from 3.

$$288x^{2}+2\times 96x^{2-1}+30x^{1-1}$$

Multiply 2 times 96.

$$288x^{2}+192x^{2-1}+30x^{1-1}$$

Subtract 1 from 2.

$$288x^{2}+192x^{1}+30x^{1-1}$$

Subtract 1 from 1.

$$288x^{2}+192x^{1}+30x^{0}$$

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

$$288x^{2}+192x+30x^{0}$$

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

$$288x^{2}+192x+30\times 1$$

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

$$288x^{2}+192x+30$$

Differentiate w.r.t. x

$$576x+192$$