Problem 10 - Calculus

Question

Find the derivatives of the functions $$ e^{1 /(1+x)} $$

Step-by-Step Solution

Answer

The derivative is \(-\frac{e^{1/(1+x)}}{(1+x)^2}\).

Step 1: Identify the function structure

The given function is \( e^{1/(1+x)} \). This is an exponential function where the exponent is a more complex function, specifically \( 1/(1+x) \). We need to use the chain rule for differentiation here.

Step 2: Apply the Chain Rule

The chain rule states that if you have a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \). Here, \( f(u) = e^u \) and \( g(x) = 1/(1+x) \). First, differentiate \( f(u) = e^u \) to get \( f'(u) = e^u \).

Step 3: Differentiate the Inner Function

Now, differentiate \( g(x) = \frac{1}{1+x} \). This can be rewritten as \( (1+x)^{-1} \). Apply the power rule for differentiation: \( \frac{d}{dx}[(1+x)^{-1}] = -1(1+x)^{-2} \) or \( -\frac{1}{(1+x)^2} \).

Step 4: Combine the Derivatives

From Step 2, \( f'(u) = e^{1/(1+x)} \). From Step 3, \( g'(x) = -\frac{1}{(1+x)^2} \). Combine these using the chain rule: \[ \frac{d}{dx}[e^{1/(1+x)}] = e^{1/(1+x)} \cdot \left(-\frac{1}{(1+x)^2}\right) \].

Step 5: Simplify the Result

The derivative can be written as: \[ \frac{d}{dx}[e^{1/(1+x)}] = -\frac{e^{1/(1+x)}}{(1+x)^2} \]. Ensure that you have simplified all components correctly.

Key Concepts

Derivatives

In calculus, the concept of a derivative represents the rate at which a function changes. To understand it better, consider a graph of a function. The derivative at any point on this graph tells us the slope of the tangent line to the function at that point.

Calculating the derivative involves applying differentiation rules to the function.

• The basic rule involves using limits to measure how a function changes as its input changes.
• For simple functions like linear ones, this is straightforward and involves basic arithmetic operations.
• For more complex functions, we use additional rules like the product, quotient, and chain rules.

Understanding how to derive a function's derivative provides important information on how rapidly or slowly the function is changing at any given point on its domain. This is fundamental in calculus and critical for understanding advanced mathematical concepts.

Chain Rule

The chain rule is a powerful tool in calculus that allows us to find the derivative of a composite function—a function made up of two or more simpler functions. The chain rule formula is expressed as: if a function is written as a composition of two functions, say \( f(g(x)) \), then its derivative \( f'(x) \) is found using \( f'(g(x)) \cdot g'(x) \).

Here's how it works:

• Identify the outer function (in this case, the exponential function \( e^u \)).
• Differentiating this gives \( e^u \) as its own derivative because the derivative of \( e^u \) remains \( e^u \).
• Identify the inner function (\( g(x) = 1/(1+x) \)).
• Differentiate the inner function using standard differentiation rules, here applying the power rule.

By using the chain rule, we combine these derivatives, multiplying the derivative of the outer function evaluated at the inner function by the derivative of the inner function. This method is invaluable for handling complex differentiations efficiently.

Exponential Function

The exponential function is one of the most important functions in mathematics, denoted as \( e^x \), where \( e \) is the base of the natural logarithms, approximately equal to 2.71828. Exponential functions are unique because their rate of growth is proportional to their current value, which means their derivative is substantially similar to the original function itself.

In calculus, understanding exponential functions and their derivatives is crucial, especially when they appear as part of a composite function, as it entails unique differentiation rules.

• For \( e^{g(x)} \), the differentiation involves using the chain rule.
• The process leverages the property that the derivative of \( e^u \) is \( e^u \) for any function \( u \).
• Thus, when differentiating functions like \( e^{1/(1+x)} \), we first differentiate the inner component and then apply the derivative rule for exponential functions.

The elegance of the exponential function in calculus lies in its simplicity and profound applicability across various scientific phenomena, from natural growth processes to complex financial models.