Problem 31 - Calculus for Biology and Medicine

Question

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=-3 e^{x^{2}+\tan x} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -6xe^{x^2 + \tan x} - 3\sec^2 xe^{x^2 + \tan x} \).

Step 1: Identify the Function Type

The function is an exponential function where the base is the constant \( e \) and the exponent is \( x^2 + \tan x \). The function is given by \( f(x) = -3e^{x^2 + \tan x} \).

Step 2: Apply the Exponential Function Derivative Rule

Recall that the derivative of \( e^u \) with respect to \( x \) is \( e^u \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \). Here, \( u = x^2 + \tan x \).

Step 3: Differentiate the Exponent

Differentiate the exponent \( x^2 + \tan x \) with respect to \( x \). This gives \( \frac{d}{dx}(x^2) = 2x \) and \( \frac{d}{dx}(\tan x) = \sec^2 x \).

Step 4: Combine the Derivatives

The derivative of the exponent \( u \) is \( \frac{du}{dx} = 2x + \sec^2 x \).

Step 5: Differentiate the Function

Using the derivative rule from Step 2, the derivative of \( f(x) = -3e^{x^2 + \tan x} \) is: \[ f'(x) = -3e^{x^2 + \tan x} \cdot (2x + \sec^2 x) \].

Step 6: Simplify the Expression

Distribute the \( -3 \) across the terms inside the parentheses: \[ f'(x) = -3(2x + \sec^2 x)e^{x^2 + \tan x} = -6xe^{x^2 + \tan x} - 3\sec^2 xe^{x^2 + \tan x} \].

Key Concepts

Exponential Function

An exponential function is a type of mathematical expression where a constant base is raised to a variable exponent. In our exercise, the base is the mathematical constant \( e \), approximately equal to 2.71828. Exponential functions are characterized by their rapid growth or decay, depending on the sign of the exponent.

• Standard form: \( e^x \)
• Our function: \( f(x) = -3e^{x^2 + \tan x} \)

Here's how it works: the function \( e^{x^2 + \tan x} \) rapidly increases or decreases based on the value of the expression in the exponent, \( x^2 + \tan x\). The negative factor, \(-3\), indicates that its direction is flipped; growths become decays and vice versa.

Derivative Rules

Derivative rules simplify the process of finding the derivative of a function, telling us the rate of change. The main rule we use here is for exponential functions:

• The derivative of \( e^u \) is \( e^u \cdot \frac{du}{dx} \) where \( u \) is a function of \( x \).

For the given function, we let \( u = x^2 + \tan x \). This means differentiating \( e^u \) requires us to find \( \frac{du}{dx} \). These rules help systematically solve problems in calculus by breaking them into manageable parts. Using these rules efficiently allows us to handle even complex expressions with ease.

Chain Rule

The chain rule is a fundamental technique in calculus used for differentiating composite functions. It states that if a function \( y = g(f(x)) \) is composed of two functions \( f(x) \) and \( g(u) \), then its derivative with respect to \( x \) is:

• \( \frac{dy}{dx} = g'(u) \cdot f'(x) \)

In our context:- We have the outer function \( e^u \) where \( u = x^2 + \tan x \).- The derivative of the outer function is multiplied by the derivative of \( u \).Using the chain rule allows us to dissect complicated expressions by focusing on one layer at a time, simplifying the differentiation process.

Differentiation Steps

Differentiating a complex function requires systematically applying calculus rules in a sequence of steps. Here is a concise breakdown of our problem:1. **Identify and express components**: We identify \( f(x) = -3e^{x^2 + \tan x} \).2. **Differentiate the inner function**: \( \frac{du}{dx} = 2x + \sec^2 x \) as the derivative of \( x^2 + \tan x \).3. **Apply the exponential function rule**: The derivative rule gives us \( -3e^{x^2 + \tan x} \cdot (2x + \sec^2 x) \).4. **Simplify the result**: Simplifying leads to \( -6xe^{x^2 + \tan x} - 3\sec^2 xe^{x^2 + \tan x} \).Systematically following these differentiation steps helps in tackling derivatives step by step, ensuring nothing is overlooked.