Question

Suppose that f(x) = e^x and g(x) = e^(cx), where c is a constant. Use mathematical induction together with the chain rule and the fact that f'(x) = e^x to prove that g^(n) = c^ne^(cx) whenever n is a positive integer.

Answer 1

To prove that g^n = c^n*e^(cx) for positive integers n, mathematical induction and the chain rule can be used. The base case is n = 1 and the inductive step assumes g^k = c^k*e^(cx). By applying the chain rule and product rule, we can differentiate g^k to get g^(k+1), which matches the desired form.

Step-by-step explanation:

To prove that g^(n) = c^n*e^(cx) for positive integers n, we will use mathematical induction and the chain rule.

Base Case: For n = 1, we have g^(1) = g' = c*e^(cx) (by definition of derivative). This matches the desired form.

Inductive Step: Assume that g^(k) = c^k*e^(cx) for some positive integer k. We want to show that g^(k+1) = c^(k+1)*e^(cx).

Using the chain rule, we have g' = (e^(cx))'. Applying the chain rule to f(x) = e^x, we find that f'(x) = e^x.

Now, using our assumption and the chain rule, we can differentiate g^(k) = c^k*e^(cx) to get g^(k+1) = (c^k*e^(cx))'. By applying the product rule, we obtain g^(k+1) = c^k*(c*e^(cx)) = c^(k+1)*e^(cx), which matches the desired form.