Solve d/dxleft(left(2x^4+2xright)^2*4x^2right)

$$\frac{d}{d x } \left( { \left(2 { x }^{ 4 } +2x \right) }^{ 2 } \times 4 { x }^{ 2 } \right)$$

Evaluate

$$160x^{9}+224x^{6}+64x^{3}$$

Steps Using Derivative Rule for Sum

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(4\left(x^{4}\right)^{2}+8x^{4}x+4x^{2}\right)\times 4x^{2})$$

To raise a power to another power, multiply the exponents. Multiply 4 and 2 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(4x^{8}+8x^{4}x+4x^{2}\right)\times 4x^{2})$$

To multiply powers of the same base, add their exponents. Add 4 and 1 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(4x^{8}+8x^{5}+4x^{2}\right)\times 4x^{2})$$

Use the distributive property to multiply 4x^{8}+8x^{5}+4x^{2} by 4.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(16x^{8}+32x^{5}+16x^{2}\right)x^{2})$$

Use the distributive property to multiply 16x^{8}+32x^{5}+16x^{2} by x^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(16x^{10}+32x^{7}+16x^{4})$$

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}.

$$10\times 16x^{10-1}+7\times 32x^{7-1}+4\times 16x^{4-1}$$

Multiply 10 times 16.

$$160x^{10-1}+7\times 32x^{7-1}+4\times 16x^{4-1}$$

Subtract 1 from 10.

$$160x^{9}+7\times 32x^{7-1}+4\times 16x^{4-1}$$

Multiply 7 times 32.

$$160x^{9}+224x^{7-1}+4\times 16x^{4-1}$$

Subtract 1 from 7.

$$160x^{9}+224x^{6}+4\times 16x^{4-1}$$

Multiply 7 times 32.

$$160x^{9}+224x^{6}+64x^{4-1}$$

Subtract 1 from 4.

$$160x^{9}+224x^{6}+64x^{3}$$

Differentiate w.r.t. x

$$1440x^{8}+1344x^{5}+192x^{2}$$