Question

Let a be a real number. We will explore the derivatives of the function f(x)=eax. By using the chain rule, we see dxd​(eax)=aeax. Recall that the second derivative of a function is the derivative of the derivative function. Similarly, the third derivative is the derivative of the second derivative. (a) What is dx2d2​(eax), the second derivative of eax ? (b) What is dx3d3​(eax), the third derivative of eax ? (c) Let n be a natural number. Make a conjecture about the nth derivative of the function f(x)=eax. That is, what is dxndn​(eax) ? This conjecture should be written as a self-contained proposition including an appropriate quantifier. 1. The Principle of Mathematical Induction 183 (d) Use mathematical induction to prove your conjecture.

Answer 1

The second derivative of the function f(x) = e^ax is a^2 * e^ax.

Step-by-step explanation:

To find the second derivative of the function f(x) = eax, we can use the chain rule. The first derivative of f(x) is f'(x) = aeax. Now, to find the second derivative, we need to take the derivative of f'(x). Using the chain rule, we get:

f''(x) = d/dx(aeax) = a * d/dx(eax) = a * a * eax = a2 * eax.

Therefore, the second derivative of f(x) = eax is f''(x) = a2 * eax.