Find the Derivative - d/dx 4x^2sin(x)

Problem

d()/d(x)*4*x2*sin(x)

Solution

1. Identify the rule needed for the derivative of a product of two functions, u(x)=4*x2 and v(x)=sin(x)

2. Apply the product rule, which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part, u=4*x2 using the power rule.

(d(4)*x2)/d(x)=8*x

4. Differentiate the second part, v=sin(x) using the trigonometric derivative rule.

d(sin(x))/d(x)=cos(x)

5. Substitute these derivatives back into the product rule formula.

(d(4)*x2*sin(x))/d(x)=4*x2*cos(x)+sin(x)*(8*x)

6. Simplify the expression by rearranging the terms.

(d(4)*x2*sin(x))/d(x)=4*x2*cos(x)+8*x*sin(x)

Final Answer

(d(4)*x2*sin(x))/d(x)=4*x2*cos(x)+8*x*sin(x)

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