What are the first and second derivatives of e^(sqrt x)?

Given the function \( f(x) = e^{\sqrt{x}} \), we want to find the first and second derivatives with respect to \( x \).

### 1. First Derivative
Let's start with the first derivative of \( f(x) \):

1. Rewrite the function in terms of the exponent as \( f(x) = e^{x^{1/2}} \).
2. Apply the chain rule, where the derivative of \( e^u \) with respect to \( u \) is \( e^u \), and then multiply by the derivative of \( u = x^{1/2} \).

#### Steps:
\[
f'(x) = e^{\sqrt{x}} \cdot \frac{d}{dx} \left( \sqrt{x} \right)
\]
Since \( \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \):
\[
f'(x) = e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}}
\]
So, the first derivative is:
\[
f'(x) = \frac{e^{\sqrt{x}}}{2\sqrt{x}}
\]

### 2. Second Derivative
Now, let’s find the second derivative \( f''(x) \).

Using the first derivative \( f'(x) = \frac{e^{\sqrt{x}}}{2\sqrt{x}} \), we apply the product rule:

#### Steps:
\[
f''(x) = \frac{d}{dx} \left( \frac{e^{\sqrt{x}}}{2\sqrt{x}} \right)
\]
Let \( g(x) = e^{\sqrt{x}} \) and \( h(x) = 2\sqrt{x} \), so:
\[
f''(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2}
\]
After substituting

Find the first and second derivatives of the exponential function e^(sqrt x) using the chain rule and product rule. This calculus problem involves key differentiation techniques, perfect for students studying advanced high school or college-level calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation