Find the derivative for function f(x) = e^x cos x with respect to x

To find the derivative of the function f(x) = e^x cos x, we will use the product rule. The product rule states that if you have two functions multiplied together, say u(x) and v(x), the derivative is given by:

Applying the Product Rule

Let:

• u(x) = e^x
• v(x) = cos x

Now, we need to find the derivatives of u and v:

• u'(x) = e^x
• v'(x) = -sin x

Calculating the Derivative

Using the product rule:

f'(x) = u'(x)v(x) + u(x)v'(x)

Substituting the values we found:

f'(x) = e^x cos x + e^x (-sin x)

Final Result

This simplifies to:

f'(x) = e^x (cos x - sin x)

Thus, the derivative of the function f(x) = e^x cos x with respect to x is f'(x) = e^x (cos x - sin x).