Question

Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=e^{2 x}$$

Short answer

Question: Express the relationship between a small change in x (dx) and the corresponding change in y (dy) for the function \(f(x) = e^{2x}\). Answer: The relationship between dx and dy for the given function is \(dy = (2e^{2x}) dx\).

Step-by-step solution

Find the derivative of the function

We need to find the derivative of the function \(f(x) = e^{2x}\). We can use the chain rule, which states that for a composite function \((g \circ h)(x) = g(h(x))\), we have: \((g \circ h)'(x) = g'(h(x)) \cdot h'(x)\) In our case, let \(g(u)=e^u\) and \(h(x)=2x\). Then, \(f(x)=g(h(x))=e^{2x}\). It's derivatives are: $$g'(u) = e^u$$ $$h'(x) = 2$$ Using the chain rule: $$f'(x) = g'(h(x)) \cdot h'(x) = e^{2x} \cdot 2$$

Express the relationship between dx and dy

Now that we have the derivative \(f'(x) = 2e^{2x}\), we can express the relationship between the small change in x and the corresponding change in y. Using \(dy = f'(x)dx\): $$dy = (2e^{2x}) dx$$ This equation tells us that the change in \(y\) is equal to 2 times the exponential function of \(2x\), multiplied by the small change in \(x\).

Key concepts

Chain Rule

The chain rule is a fundamental principle in differential calculus used to calculate the derivative of a composite function. When one function is applied within another, the chain rule enables us to differentiate the result with respect to its independent variable.

Imagine two functions, for example, where one function 'h' is nested inside another function 'g'. The derivative of the composite function, which we might write as 'f(x) = g(h(x))', is found by taking the derivative of the outer function 'g' with respect to its inner function 'h(x)', and then multiplying this by the derivative of the inner function 'h' with respect to 'x'. This process effectively 'chains' the derivatives together, hence its name.

Mathematically put, if you have functions such that 'f(x) = g(h(x))', the derivative of 'f' with respect to 'x' is expressed as:\(f'(x) = g'(h(x)) \times h'(x)\).
In terms of incremental changes, this principle allows us to understand how a small change in 'x' can affect 'y' through the interplay of the inner and outer functions.

Derivatives

Derivatives represent the rate at which one variable changes with respect to another. In terms of functions, the derivative of a function at a certain point gives the slope of the tangent line to the function's graph at that point. The process of finding a derivative is known as differentiation.

For the function \(f(x)\), the derivative is often denoted as \(f'(x)\) or \(\frac{df}{dx}\), where 'df' signifies a tiny change in function 'f', and 'dx' signifies a tiny change in variable 'x'. The derivative tells us how 'f' changes for a small change in 'x', which is the very essence of infinitesimal calculus. Knowing the derivative is crucial when trying to predict and understand the behavior of functions, as it provides information about rates of change and can help in solving problems involving motion, growth, and optimization.

Exponential Functions

Exponential functions are powerful mathematical expressions that involve the constant 'e' (approximately equal to 2.71828), raised to a variable power. An exponential function is generally expressed as \(f(x) = e^{kx}\), where 'k' is a constant. These functions are known for their unique property: the rate of growth of the function is directly proportional to the value of the function itself.

What makes exponential functions particularly interesting is how they model real-world phenomena such as population growth, radioactive decay, and compound interest. As 'x' increases or decreases, the function rises or falls at a rate that is constantly changing, which is why the concept of the derivative is especially important for these functions. The derivative of an exponential function is directly related to the function itself, making the process of finding changes in the function over time more complex yet fascinating.

Infinitesimal Changes

Infinitesimal changes refer to changes that are extremely small or minute—in terms of calculus, we consider these to be infinitely small. When working with functions in calculus, we often need to understand the behavior of these functions as the input variables experience these infinitesimal changes.

This concept is the cornerstone of differential calculus and allows mathematicians to make sense of changes that would otherwise be too small to reason with conventional means. By considering the ratio of the infinitesimal change in the function's output (denoted 'dy') to the infinitesimal change in the input variable (denoted 'dx'), we get the derivative. Understanding the derivative as this ratio is key in explaining not just the instantaneous rate of change, but also how different variables influence each other at the margin, which is paramount in fields like economics, physics, and engineering.