Question

Find the differential \(d y\) of the given function. $$ y=x \sin x $$

Short answer

The differential \(d y\) of the function \(y = x \sin x\) is \(d y = [\sin x + x \cos x] dx\).

Step-by-step solution

Identify the functions

In this case, we have \(u = x\) and \(v = \sin x\). These are the two functions that make up \(y\).

Compute the derivatives of u and v

The derivative of \(u = x\) is \(u' = 1\). The derivative of \(v = \sin x\) is \(v' = \cos x\).

Apply the product rule

The product rule states that \((u v)' = u' v + u v'\). Applying this to our functions, we have \(y' = u' v + u v' = (1)(\sin x) + (x)(\cos x)\).

Compute the differential

The differential \(d y\) is then the derivative \(y'\) multiplied by \(dx\), so \(d y = [(1)(\sin x) + (x)(\cos x)] dx\).

Key concepts

Product Rule

In differential calculus, the product rule is a critical tool for taking the derivative of a product of two functions. Imagine you have two functions, like dancers in a choreographed routine. The product rule lets you break down their complex movements (the product of their functions) into simpler steps, so you can understand each dancer's part.

Here’s how it works: If you have two functions, let’s call them u and v, and you want to take the derivative of their product u*v, the product rule says that the derivative is u'v + uv'. To breakdown it down, you first find the derivatives of u and v separately, denoted as u' and v'. Then, you cross-multiply the original function with the derivative of the other function, adding those results together. This results in a new function that is the derivative of the original product.

For instance, in our exercise, calculating the derivative of x * sin(x) using the product rule gives us two parts:

• The derivative of x, which is 1, times sin(x),
• Plus, x times the derivative of sin(x), which is cos(x).

The final expression (1)(sin(x)) + (x)(cos(x)) is the derivative of the product x*sin(x) according to the product rule.

Derivative

The term derivative might sound complicated, but it’s really just a fancy word for a concept you’re already familiar with: the rate of change. Whether it's the speed of a car or the slope of a hill, the derivative helps us quantify how one thing changes in relation to another.

Mathematically, when we have a function y=f(x), the derivative tells us how y changes as x changes. It’s like looking through a microscope at tiny increments of change to predict what's going to happen next. The notation f'(x) or df/dx represents the derivative of f with respect to x. Calculating a derivative is like finding the slope of the tangent line to the curve of f at any point.

In the exercise provided, the derivative of the function y = x*sin(x) tells us how y changes as x changes. After applying the product rule, we find that the derivative y' or dy/dx is a combination of sine and cosine functions which shows the rate of change of the initial function.

Trigonometric Functions

Trigonometric functions are the sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent). These functions play a starring role in both geometry and calculus. They are essential in describing the relationships between the angles and sides of triangles, but they also extend to describe patterns of change in periodic phenomena.

Sin(x) and cos(x), specifically, are like a pair of dancers who move in sync – one reaches its high as the other reaches its low. Their derivatives, which represent the rate at which their values change, are interconnected: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x).

In our exercise, we see these functions in action. The function y = x*sin(x) combines a linear function x and a trigonometric function sin(x). The beauty of trigonometry in calculus is evident when we apply the product rule, as it involves taking the derivatives of these trigonometric functions to find the rate of change of y. Understanding the behaviour of these functions and their derivatives is essential for solving a wide array of problems in calculus.