Question

Let f(x) = -11xsin(x)cos(x). Evaluate f'(x) at x = π.

Answer 1

To find the derivative of -11xsin(x)cos(x), use the product rule and the chain rule. Evaluate the derivative at x = π to find the exact value.

Step-by-step explanation:

To find the derivative of the function f(x) = -11xsin(x)cos(x), we need to use the product rule and the chain rule. The product rule states (fg)' = f'g + fg', where f' and g' are the derivatives of f and g respectively. In this case, f(x) = -11x and g(x) = sin(x)cos(x).

First, let's find the derivative of f(x). Since f(x) = -11x, the derivative of f(x) with respect to x is f'(x) = -11.

Next, let's find the derivative of g(x). Using the chain rule, the derivative of g(x) = sin(x)cos(x) is g'(x) = cos^2(x) - sin^2(x).

Now, we can apply the product rule to find the derivative of f(x)g(x). f'(x)g(x) + f(x)g'(x) = (-11)(sin(x)cos(x)) + (-11x)(cos^2(x) - sin^2(x)).

Finally, evaluate the derivative at x = π to find the exact value.