Solve d/dx4x(-2x^2cos(x^2)-2sin(x^2))

$$\frac { d } { d x } 4 x ( - 2 x ^ { 2 } \cos ( x ^ { 2 } ) - 2 \sin ( x ^ { 2 } ) )$$

Evaluate

$$8\left(2x^{4}\sin(x^{2})-5x^{2}\cos(x^{2})-\sin(x^{2})\right)$$

Steps Using Derivative Rule for Sum

Multiply -1 and 2 to get -2.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x\left(-2x^{2}\cos(x^{2})-2\sin(x^{2})\right))$$

Use the distributive property to multiply 4x by -2x^{2}\cos(x^{2})-2\sin(x^{2}).

$$\frac{\mathrm{d}}{\mathrm{d}x}(-8x^{3}\cos(x^{2})-8\sin(x^{2})x)$$

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}.

$$3\left(-8\cos(x^{2})\right)x^{3-1}+\left(-8\sin(x^{2})\right)x^{1-1}$$

Multiply 3 times -8\cos(x^{2}).

$$\left(-24\cos(x^{2})\right)x^{3-1}+\left(-8\sin(x^{2})\right)x^{1-1}$$

Subtract 1 from 3.

$$\left(-24\cos(x^{2})\right)x^{2}+\left(-8\sin(x^{2})\right)x^{1-1}$$

Subtract 1 from 1.

$$\left(-24\cos(x^{2})\right)x^{2}+\left(-8\sin(x^{2})\right)x^{0}$$

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

$$\left(-24\cos(x^{2})\right)x^{2}+\left(-8\sin(x^{2})\right)\times 1$$

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

$$\left(-24\cos(x^{2})\right)x^{2}-8\sin(x^{2})$$

Differentiate w.r.t. x

$$16x\left(2x^{4}\cos(x^{2})+9x^{2}\sin(x^{2})-6\cos(x^{2})\right)$$