Solve d/dx4(x^2+x)sine^x

$$\frac { d } { d x } 4 ( x ^ { 2 } + x ) \sin e ^ { x }$$

Evaluate

$$4\left(x\left(x+1\right)e^{x}\cos(e^{x})+\left(x+1\right)\sin(e^{x})+x\sin(e^{x})\right)$$

Steps Using Derivative Rule for Sum

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

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

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{2}\sin(e^{x})+4x\sin(e^{x}))$$

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}.

$$2\times 4\sin(e^{x})x^{2-1}+4\sin(e^{x})x^{1-1}$$

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

$$8\sin(e^{x})x^{2-1}+4\sin(e^{x})x^{1-1}$$

Subtract 1 from 2.

$$8\sin(e^{x})x^{1}+4\sin(e^{x})x^{1-1}$$

Subtract 1 from 1.

$$8\sin(e^{x})x^{1}+4\sin(e^{x})x^{0}$$

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

$$8\sin(e^{x})x+4\sin(e^{x})x^{0}$$

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

$$8\sin(e^{x})x+4\sin(e^{x})\times 1$$

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

$$8\sin(e^{x})x+4\sin(e^{x})$$

Differentiate w.r.t. x

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