Solve =d/dx(x^{2}(3x^2+4)^2(3-x^3)^4)

$$= \frac { d } { d x } ( x ^ { 2 } ( 3 x ^ { 2 } + 4 ) ^ { 2 } ( 3 - x ^ { 3 } ) ^ { 4 } )$$

Evaluate

$$2x\left(3x^{2}+4\right)\left(12+27x^{2}-28x^{3}-27x^{5}\right)\left(3-x^{3}\right)^{3}$$

Steps Using Derivative Rule for Sum

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x^{2}+4\right)^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}\left(9\left(x^{2}\right)^{2}+24x^{2}+16\right)\left(3-x^{3}\right)^{4})$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{2}\left(9x^{4}+24x^{2}+16\right)\left(3-x^{3}\right)^{4})$$

Use the distributive property to multiply x^{2} by 9x^{4}+24x^{2}+16.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(9x^{6}+24x^{4}+16x^{2}\right)\left(3-x^{3}\right)^{4})$$

Use the distributive property to multiply 9x^{6}+24x^{4}+16x^{2} by \left(3-x^{3}\right)^{4}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(9x^{6}\left(3-x^{3}\right)^{4}+24x^{4}\left(3-x^{3}\right)^{4}+16x^{2}\left(3-x^{3}\right)^{4})$$

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}.

$$6\times 9\left(3-x^{3}\right)^{4}x^{6-1}+4\times 24\left(3-x^{3}\right)^{4}x^{4-1}+2\times 16\left(3-x^{3}\right)^{4}x^{2-1}$$

Multiply 6 times 9\left(3-x^{3}\right)^{4}.

$$54\left(3-x^{3}\right)^{4}x^{6-1}+4\times 24\left(3-x^{3}\right)^{4}x^{4-1}+2\times 16\left(3-x^{3}\right)^{4}x^{2-1}$$

Subtract 1 from 6.

$$54\left(3-x^{3}\right)^{4}x^{5}+4\times 24\left(3-x^{3}\right)^{4}x^{4-1}+2\times 16\left(3-x^{3}\right)^{4}x^{2-1}$$

Multiply 4 times 24\left(3-x^{3}\right)^{4}.

$$54\left(3-x^{3}\right)^{4}x^{5}+96\left(3-x^{3}\right)^{4}x^{4-1}+2\times 16\left(3-x^{3}\right)^{4}x^{2-1}$$

Subtract 1 from 4.

$$54\left(3-x^{3}\right)^{4}x^{5}+96\left(3-x^{3}\right)^{4}x^{3}+2\times 16\left(3-x^{3}\right)^{4}x^{2-1}$$

Multiply 4 times 24\left(3-x^{3}\right)^{4}.

$$54\left(3-x^{3}\right)^{4}x^{5}+96\left(3-x^{3}\right)^{4}x^{3}+32\left(3-x^{3}\right)^{4}x^{2-1}$$

Subtract 1 from 2.

$$54\left(3-x^{3}\right)^{4}x^{5}+96\left(3-x^{3}\right)^{4}x^{3}+32\left(3-x^{3}\right)^{4}x^{1}$$

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

$$54\left(3-x^{3}\right)^{4}x^{5}+96\left(3-x^{3}\right)^{4}x^{3}+32\left(3-x^{3}\right)^{4}x$$

Differentiate w.r.t. x

$$2\left(x^{3}-3\right)^{2}\left(1377x^{10}+2880x^{8}-3078x^{7}+1456x^{6}-5184x^{5}+1215x^{4}-1824x^{3}+1296x^{2}+144\right)$$