Question

In Exercises, find the derivative of the function. $$ y=e^{-x^{2}} $$

Short answer

The derivative of \( \ln (x^{2} + 3) \) is \(\frac{2x}{x^{2} + 3}\)

Step-by-step solution

Differentiate Primary Function

Begin by applying the derivative of the natural logarithm function, which is \( 1/x \). However, keep in mind that here, 'x' is the entire content inside the parentheses - namely, \(x^2 + 3\). This gives us: \( (1/(x^2 + 3)) \)

Apply Chain Rule

Then apply the chain rule, which tells us to take the derivative of the inner function (the inside of the parentheses). The derivative of \(x^2 + 3\) is \(2x\). Multiply this with the result from step 1: \( 2x * (1/(x^2 + 3))\) . This may also be written as \( 2x/(x^2 + 3) \).

Key concepts

Chain Rule

The chain rule is a fundamental tool in calculus, particularly when it comes to differentiation. Think of the chain rule as a way to peel back layers like an onion. When you have a function that is composed of multiple other functions, such as a function within a function, you use the chain rule to take the derivative of each layer one by one.

Here’s how it works in essence: Suppose you have two functions, let’s call them u and v, where u is a function of x and v is a function of u. When you need to find the derivative of v(u(x)), the chain rule states that you should multiply the derivative of the outer function v (with respect to u) by the derivative of the inner function u (with respect to x). In mathematical terms:
\[\begin{equation}\frac{d}{dx}[v(u(x))] = v'(u(x)) \times u'(x).\end{equation}\]
Applying this to our example problem, where the outer function is the natural logarithm of u, and u is the quadratic expression x^2 + 3, using the chain rule is pivotal to correctly find the derivative.

Natural Logarithm Differentiation

When it comes to the natural logarithm, usually denoted as ln, its differentiation may appear daunting at first, but it follows a simple rule. The derivative of ln(x) with respect to x is \[\begin{equation}1/x\end{equation}\] provided x > 0. This rule is essential because natural logarithms frequently appear in various forms within calculus.

To demonstrate, let’s apply this to the expression \[\begin{equation}ln(x^2 + 3)\end{equation}\]from the given exercise. Initially, we identify x^2 + 3 as the inner function, and apply the differentiation rule of the natural logarithm to it, treating x^2 + 3 as our 'x'. Thus, we obtain the derivative of \[\begin{equation}1/(x^2 + 3)\end{equation}\]as the first part of finding our overall derivative.

Calculus Step by Step Solutions

Understanding calculus requires not only knowledge of the rules of differentiation and integration but also the ability to apply these rules in a step by step process to solve problems. This is why step by step solutions are so valuable for learning and comprehending the material; they provide a logical sequence that can be followed, allowing learners to not only get the correct answer but understand how it was achieved.

For our exercise, we followed a two-step process.

First, identify the primary function to differentiate

We recognized that we had a natural logarithm function containing another function.

Second, apply the chain rule

We then differentiated the inner function, in this case x^2 + 3, to get \[\begin{equation}2x.\end{equation}\]Finally, we multiplied this derivative with the derivative of the outer function (the natural logarithm part) to arrive at the full derivative: \[\begin{equation}2x/(x^2 + 3).\end{equation}\]Through this systematic approach, the step by step solution not only provides the answer but also reinforces the understanding of differentiation techniques, such as the chain rule and the differentiation of natural logarithms.