Problem 7 - Thomas Calculus in SI Units

Question

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta,\) as appropriate. $$y=e^{5-7 x}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dx} = -7e^{5-7x} \).

Step 1: Identify the Function and the Variable

The function is given as \( y = e^{5-7x} \). The task is to find the derivative of \( y \) with respect to \( x \).

Step 2: Recall the Derivative Rule for Exponential Functions

For a function \( y = e^{u(x)} \), the derivative is given by \( \frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx} \) where \( u(x) \) is a differentiable function of \( x \).

Step 3: Differentiate the Exponent

Identify \( u(x) = 5-7x \). The derivative of \( u(x) \) with respect to \( x \) is \( \frac{du}{dx} = -7 \).

Step 4: Apply the Exponential Derivative Rule

Substitute \( u(x) = 5-7x \) and \( \frac{du}{dx} = -7 \) into the derivative formula: \( \frac{dy}{dx} = e^{5-7x} \cdot (-7) = -7e^{5-7x} \).

Step 5: Write the Final Answer

Thus, the derivative of \( y = e^{5-7x} \) with respect to \( x \) is \( \frac{dy}{dx} = -7e^{5-7x} \).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. The most widely used exponential function in calculus is the natural exponential function, expressed as \(e^x\). Here, \(e\) is the base of the natural logarithm, approximately equal to 2.71828. These functions are powerful because they model many real-world situations, like population growth or radioactive decay.

• Natural exponential functions have the distinctive property that their rate of change is proportional to the function's current value.
• The equation \(y = e^{5-7x}\), as seen in the exercise, represents a transformation of the basic exponential function using the exponent \(5-7x\).
• This transformation indicates rapid change and is essential for solving derivative exercises.

Differentiation Rules

Differentiation is a fundamental concept in calculus used to find the rate at which a quantity changes. Several rules aid in differentiating various types of functions, with exponential functions having their unique approach.

• The derivative of an exponential function \(e^{u(x)}\) with respect to \(x\) is \(e^{u(x)} \cdot \frac{du}{dx}\).
• This rule involves finding the derivative of the exponent \(u(x)\) alongside the exponential function itself.
• In the exercise, \(\frac{du}{dx} = -7\) since \(u(x) = 5 - 7x\). This makes the derivative of \(y\) equal to \(-7e^{5-7x}\).

By applying these rules, you can easily differentiate exponential functions and solve a variety of calculus problems.

Calculus

Calculus is the branch of mathematics that studies continuous change, encompassing derivatives and integrals. It provides the tools needed to analyze and model dynamic systems found in science and engineering.

• At its core, calculus involves understanding how quantities change over time, and derivatives are used to calculate these rates of change.
• In this exercise, the goal was to find the derivative of the function \(y = e^{5-7x}\), which signifies the function's instantaneous rate of change with respect to \(x\).
• This concept is applicable in numerous fields, including physics, economics, and biology, for modeling changing systems.

Understanding calculus and its rules, like differentiation, equips you with the skills to solve complex problems and uncover relationships in dynamic systems.