The chain rule is a rule in calculus that allows you to find the derivative of a composite function. It states that the derivative of a composition of two functions is equal to the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.
In this scenario, we can apply the power rule by first rewriting the function as cos(e^(-θ^2)) and then using the fact that the derivative of cos(x) is -sin(x). This gives us -sin(e^(-θ^2)).
Step 1: Rewrite the function as cos(e^(-θ^2)).
Step 2: Identify the inner function, in this case e^(-θ^2).
Step 3: Find the derivative of the outer function, which is -sin(x).
Step 4: Find the derivative of the inner function, which is -2θe^(-θ^2).
Step 5: Multiply the two derivatives together, giving us -sin(e^(-θ^2)) * -2θe^(-θ^2).
Step 6: Simplify the expression to get the final answer of 2θsin(e^(-θ^2)).
In addition to the power rule and the chain rule, we also need to apply the exponential rule when finding the derivative of e^(-θ^2). This rule states that the derivative of e^x is e^x, so the derivative of e^(-θ^2) is -2θe^(-θ^2).
Yes, the chain rule can be applied to any composite function as long as the derivatives of the individual functions are known. It is a fundamental rule in calculus and is used frequently in finding derivatives of more complex functions.