Find the Derivative - d/dx xcos(x)

Problem

(d(x)*cos(x))/d(x)

Solution

1. Identify the rule needed for the derivative. Since the expression x*cos(x) is a product of two functions, x and cos(x) the product rule must be used.

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

3. Differentiate the individual components. The derivative of x is 1 and the derivative of cos(x) is −sin(x)

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

(d(x)*cos(x))/d(x)=x⋅d(cos(x))/d(x)+cos(x)⋅d(x)/d(x)

5. Simplify the resulting expression by performing the multiplication.

(d(x)*cos(x))/d(x)=x*(−sin(x))+cos(x)*(1)

6. Rearrange the terms for the final result.

(d(x)*cos(x))/d(x)=−x*sin(x)+cos(x)

Final Answer

(d(x)*cos(x))/d(x)=cos(x)−x*sin(x)

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