Problem 25 - Calculus for Biology and Medicine

Question

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=\exp [x-\sin x] $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \exp[x - \sin x] \cdot (1 - \cos x) \).

Step 1: Identify the Function

The given function is \( f(x) = \exp[x - \sin x] \). Our task is to differentiate this function with respect to \( x \).

Step 2: Recognize the Chain Rule

To differentiate \( \exp[u] \) where \( u = x - \sin x \), we use the chain rule. The chain rule states \( \frac{d}{dx} \exp[u] = \exp[u] \cdot \frac{du}{dx} \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( u = x - \sin x \) with respect to \( x \). Using basic differentiation rules: \( \frac{du}{dx} = \frac{d}{dx}(x) - \frac{d}{dx}(\sin x) = 1 - \cos x \).

Step 4: Apply the Chain Rule

Substitute \( u = x - \sin x \) and \( \frac{du}{dx} = 1 - \cos x \) into the chain rule expression: \( \frac{d}{dx} \exp[x - \sin x] = \exp[x - \sin x] \cdot (1 - \cos x) \).

Step 5: Simplify the Derivative

The derivative of the function is \( f'(x) = \exp[x - \sin x] \cdot (1 - \cos x) \). This is the final differentiated expression.

Key Concepts

Chain Rule

The chain rule is a fundamental concept in calculus differentiation, used to differentiate compositions of functions. It is exceptionally handy when you have functions inside other functions, such as in the given problem with the exponential function nested inside. The idea is straightforward: you differentiate the outer function and multiply it by the derivative of the inner function.

To apply the chain rule to the function \( f(x) = \exp[x - \sin x] \), consider \( u = x - \sin x \) as the inner function and \( \exp[u] \) as the outer function. The chain rule then tells us:

• Differentiating the outer function: \( \frac{d}{du} \exp[u] = \exp[u] \)
• Differentiating the inner function, \( u = x - \sin x \), we find \( \frac{du}{dx} = 1 - \cos x \)

By multiplying these two results, we apply the chain rule, giving us the complete derivative. This process greatly simplifies tackling complex functions.

Differentiation Techniques

Differentiation techniques are the methods used to find a derivative, which represents the rate at which a function changes. When dealing with different types of functions, such as polynomials, trigonometric functions, or exponential functions, we can employ various techniques. Here are the main methods:

• Basic differentiation rules include the derivative of a constant (which is 0) and the derivative of \( x^n \) (which is \( nx^{n-1} \)).
• Product and quotient rules are applicable when dealing with products or ratios of functions.
• Chain rule, as discussed earlier, is crucial for composite functions.

For the problem at hand, we mainly used basic rules for the inner function and the chain rule to connect it with the outer exponential function. By understanding and correctly applying these techniques, you can effectively tackle a wide range of differentiation problems.

Exponential Function Differentiation

Exponential functions have a distinctive feature: their rate of change is proportional to their value. This makes them drastically different from other types of functions, such as polynomials. The most recognizable exponential function is \( \, e^x \), where \( e \) is the mathematical constant approximately equal to 2.718.

When differentiating exponential functions, the derivative of \( \exp[u] \) with respect to \( u \) is simply \( \exp[u] \). This property makes them straightforward to handle when combined with the chain rule. For example, in our problem, the derivative of \( \exp[x - \sin x] \) involves multiplying \( \exp[x - \sin x] \) by the derivative of its inner function \( 1 - \cos x \), as determined using the chain rule.

This characteristic of retaining its exponential form even after differentiation is why exponential functions are widely used in models of real-world exponential growth or decay. Understanding this unique property can provide insight into the behavior of these functions and their applications across various fields.