Find the Derivative - d/dx e^(xcos(x))

Problem

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

Solution

1. Identify the outer function as eu and the inner function as u=x*cos(x)

2. Apply the chain rule, which states that d(eu)/d(x)=eu⋅d(u)/d(x)

3. Apply the product rule to find the derivative of the inner function x*cos(x) using the formula (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

4. Differentiate the individual terms: d(x)/d(x)=1 and d(cos(x))/d(x)=−sin(x)

5. Substitute the result of the product rule back into the chain rule expression.

6. Simplify the final expression by combining terms inside the parentheses.

Final Answer

d(e(x*cos(x)))/d(x)=e(x*cos(x))*(cos(x)−x*sin(x))

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