Question

In Exercises \(39-68,\) find the derivative of the function. \(y=\cos 3 x\)

Short answer

The derivative of the function \(y= \cos 3x\) is \(-3\sin(3x)\).

Step-by-step solution

Identify the Outside and Inside Function

The outside function here is the cosine function and the inside function is \(3x\). This is because the \(3x\) is 'inside' the cosine function in the setting of this problem.

Derivative of the Inside Function

First, we find the derivative of the inside function. Here, the inside function is \(3x\). The derivative of \(3x\) with respect to \(x\) is \(3\). So \(\frac{d}{dx}(3x) = 3\).

Derivative of the Outside Function

Next, we find the derivative of the outside function, which is \(\cos(x)\). The derivative of \(\cos(x)\) with respect to \(x\) is \(-\sin(x)\). So, \(\frac{d}{dx}(\cos(x)) = -\sin(x)\).

Applying the Chain Rule

The chain rule states that the derivative of a composed function is the derivative of the outside function times the derivative of the inside function. The outside function here is \(\cos(x)\) and its derivative is \(-\sin(x)\). The inside function is \(3x\) and its derivative is \(3\). Now we apply the chain rule by multiplying the derivative of the outside function by the derivative of the inside function, and replacing \(x\) in the \(-\sin(x)\) by inside function, i.e., \(3x\). Doing this give us the derivative of the given function as follows: \(-\sin(3x) \times 3\).

Key concepts

Derivative of the Cosine Function

Understanding how to take the derivative of the cosine function is crucial when dealing with trigonometric functions in calculus. The cosine function, which is usually written as \(\cos(x)\), varies between -1 and 1 as its input, \(x\), moves through the angles of a circle.

The derivative of \(\cos(x)\) is found using the rules of differentiation. It's known that the rate of change of the cosine function with respect to \(x\) is \(\sin(x)\), but it is negative because the cosine function decreases as \(x\) increases in the first quadrant of the unit circle. This is succinctly expressed as \(\frac{d}{dx}\cos(x) = -\sin(x)\).

In the context of the exercise, when we have \(\cos(3x)\), this involves not only the cosine function but also a composition with the linear function \(3x\), which precipitates the use of the chain rule for correct differentiation.

Composition of Functions

Composition of functions is a core concept in calculus, referring to the combination of two functions where the output of one function becomes the input of another. If we have two functions, \(f(x)\) and \(g(x)\), the composition of \(f\) and \(g\), denoted by \(f(g(x))\), is the function obtained by applying \(g\) first and then applying \(f\) to the result.

Example of Composition

As an illustration, if \(f(x) = x^2\) and \(g(x) = 3x\), the composition \(f(g(x))\), which can be read as 'f of g of x', is \(f(3x) = (3x)^2\). When composing functions, it's crucial to keep the order straight, as \(f(g(x))\) can yield a different result from \(g(f(x))\).

The concept of composition is essential when applying the chain rule, as it involves taking the derivative of a composed function. In our case, the composed function is \(\cos(3x)\), where the composition is between the cosine function and the linear function \(3x\).

Applying the Chain Rule

The chain rule is a powerful tool in calculus used to differentiate composed functions. It essentially allows us to break down the differentiation process into simpler steps by taking the derivatives of the 'outer' and 'inner' functions separately, and then multiplying them together.

Following the steps from the textbook solution, the inside function is \(3x\) and the outside function is \(\cos(x)\). The derivative of the inside function, \(3x\), is \(3\), and the derivative of the outside function, \(\cos(x)\), is \(\-\sin(x)\). Applying the chain rule, we multiply the derivative of the outside function by the derivative of the inside function:

\(\frac{d}{dx}\cos(3x) = \frac{d}{dx}\cos(x) \cdot \frac{d}{dx}(3x) = -\sin(x) \cdot 3\).

It's essential to remember that when we apply the derivative of the outside function, we do so by replacing \(x\) with the inside function, resulting in \(\-\sin(3x)\). Multiplying this by the derivative of the inside function gives us the final answer: \(\-3\sin(3x)\).

The chain rule provides a systematic way to handle derivatives of functions that are not just simple polynomials or single trigonometric functions, but rather a combination of several functions, which is a common occurrence in mathematics.