Solve d/dx[2x(-2x^2sinx^2+3cos(x^2))]

$$\frac { d } { d x } [ 2 x ( - 2 x ^ { 2 } \sin x ^ { 2 } + 3 \cos ( x ^ { 2 } ) ) ]$$

Evaluate

$$2\left(-4x^{4}\cos(x^{2})-12x^{2}\sin(x^{2})+3\cos(x^{2})\right)$$

Steps Using Derivative Rule for Sum

Use the distributive property to multiply 2x by -2x^{2}\sin(x^{2})+3\cos(x^{2}).

$$\frac{\mathrm{d}}{\mathrm{d}x}(-4\sin(x^{2})x^{3}+6x\cos(x^{2}))$$

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}.

$$3\left(-4\sin(x^{2})\right)x^{3-1}+6\cos(x^{2})x^{1-1}$$

Multiply 3 times -4\sin(x^{2}).

$$\left(-12\sin(x^{2})\right)x^{3-1}+6\cos(x^{2})x^{1-1}$$

Subtract 1 from 3.

$$\left(-12\sin(x^{2})\right)x^{2}+6\cos(x^{2})x^{1-1}$$

Subtract 1 from 1.

$$\left(-12\sin(x^{2})\right)x^{2}+6\cos(x^{2})x^{0}$$

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

$$\left(-12\sin(x^{2})\right)x^{2}+6\cos(x^{2})\times 1$$

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

$$\left(-12\sin(x^{2})\right)x^{2}+6\cos(x^{2})$$

Differentiate w.r.t. x

$$4x\left(4x^{4}\sin(x^{2})-20x^{2}\cos(x^{2})-15\sin(x^{2})\right)$$