Problem 1 - Calculus

Question

Find the derivatives of the functions $$ 7 e^{7 x} $$

Step-by-Step Solution

Answer

The derivative is \( 49e^{7x} \).

Step 1: Identify the function form

The function given is \( f(x) = 7e^{7x} \). It is a constant multiplied by an exponential function. The form of this function is \( ae^{bx} \), where \( a = 7 \) and \( b = 7 \).

Step 2: Recall the derivative formula for exponential functions

The derivative of an exponential function of the form \( ae^{bx} \) is given by \( f'(x) = abe^{bx} \). This formula comes from the chain rule of differentiation applied to exponential functions.

Step 3: Apply the derivative formula

Now, apply the formula to find the derivative of the function. Substitute \( a = 7 \) and \( b = 7 \) into the formula: \( f'(x) = 7 \cdot 7 e^{7x} = 49e^{7x} \).

Step 4: Simplify the derivative

The expression \( 49e^{7x} \) is already simplified. This is the derivative of the original function.

Key Concepts

Exponential Functions

Exponential functions have a unique growth pattern that increases rapidly as the variable becomes larger. They are usually expressed in the form of \( e^{x} \), where \( e \approx 2.71828 \) is a constant called Euler’s number. The function given in the exercise is \( 7e^{7x} \), which is a typical exponential function scaled by a constant (7, in this case) and has \( 7x \) in the exponent.
This specific structure \( ae^{bx} \) defines an exponential function where:

• \( a \) is a constant that affects the function's vertical stretch, determining how "tall" the function graph appears. In this exercise, \( a = 7 \).
• \( b \) in the exponent indicates the rate of growth; the larger the \( b \), the faster the function rises. Here, \( b = 7 \).

The property that makes exponential functions notable in calculus is how their rate of increase is directly proportional to their current value, translating to a consistent structure in their derivative.

Differentiation

Differentiation is an essential part of calculus that allows us to find the rate at which a function is changing. This is called a derivative, and it provides invaluable insight into the behavior of functions like rates of growth or change, which is crucial for understanding exponential functions.
When differentiating exponential functions of the form \( ae^{bx} \), the process transforms these functions into another exponential function governed by a simple rule. The derivative \( f'(x) \) has the form \( abe^{bx} \), where:

• The constant \( a \) remains as is.
• The derivation introduces a new term, \( b \), multiplying the original exponential function.

This results in the derivative reflecting the intrinsic ratio of change of the exponential function, maintaining its form while scaling it according to \( b \). As shown in the solution, \( 7e^{7x} \) leads to a derivative of \( 49e^{7x} \), marking the exponential behavior in both the function and its rate of change.

Chain Rule

The chain rule is pivotal when differentiating composite functions. It's especially handy for exponential functions where there's more to the expression than meets the eye. The chain rule helps us differentiate functions written as one function inside another, like \( e^{7x} \) in this exercise.
The chain rule states that if you have a function \( g(h(x)) \), its derivative is the derivative of \( g \) with respect to \( h \) times the derivative of \( h \) with respect to \( x \). It is represented mathematically as:

• \( (g(h(x)))' = g'(h(x)) \cdot h'(x) \).

For \( e^{7x} \), consider \( g(u) = e^u \) and \( h(x) = 7x \). The derivative of \( g \) concerning \( u \) is \( e^u \), and the derivative of \( h(x) \) concerning \( x \) is 7. Therefore, the derivative becomes \( 7e^{7x} \) multiplied by 7, resulting in the complete derivative: \( 49e^{7x} \), showcasing the power of the chain rule in simplifying derivatives of exponential forms.