Question

Find the derivative of the following functions. $$y=\cos ^{2} x$$

Short answer

Answer: The derivative of the function $$y = \cos^2 x$$ with respect to $$x$$ is $$\frac{dy}{dx} = -2\cos x \sin x$$.

Step-by-step solution

Identify the outer and inner functions

First, identify the outer function ($$u^2$$) and the inner function ($$u = \cos x$$). We are given the function: $$y = \cos^2 x$$. Rewrite this function in terms of $$u$$: $$y = u^2$$ Where $$u = \cos x$$.

Differentiate the outer function

Now, differentiate the outer function $$y = u^2$$ with respect to $$u$$: $$\frac{dy}{du} = 2u$$.

Differentiate the inner function

Next, differentiate the inner function $$u = \cos x$$ with respect to $$x$$: $$\frac{du}{dx} = -\sin x$$.

Apply the chain rule

Now, apply the chain rule by multiplying the derivatives of the outer and inner functions. In this case, we multiply $$\frac{dy}{du}$$ by $$\frac{du}{dx}$$ to obtain $$\frac{dy}{dx}$$: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot (-\sin x)$$.

Substitute the inner function back in

Finally, substitute $$u = \cos x$$ back into the expression for the derivative: $$\frac{dy}{dx} = 2(\cos x) \cdot (-\sin x) = -2\cos x \sin x$$. Therefore, the derivative of the function $$y = \cos^2 x$$ with respect to $$x$$ is: $$\frac{dy}{dx} = -2\cos x \sin x$$.

Key concepts

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. A composite function is one where you have a function inside another function. In this context, for example, when dealing with \(y = \cos^2 x\), the function \(\cos x\) is "inside" the function \(u^2\).
To apply the chain rule, you must first identify the inner and outer functions. Here:

• The outer function is \(y = u^2\)
• The inner function is \(u = \cos x\)

Next, you differentiate each function separately. Differentiate \(y = u^2\) to get \( \frac{dy}{du} = 2u\) and differentiate \(u = \cos x\) to get \( \frac{du}{dx} = -\sin x\). Finally, combine these derivatives using the chain rule:
\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 2u \cdot (-\sin x)\] This gives the overall derivative of the composite function.

Trigonometric Functions

Trigonometric functions, like cosine and sine, are essential in calculus and many applications of mathematics. The trigonometric function involved in the exercise \(y = \cos^2 x\) is the cosine function.
Cosine function is usually denoted as \(\cos x\), which gives the ratio of the adjacent side to the hypotenuse in a right triangle. Its derivative is important in solving calculus problems: \( \frac{d}{dx}(\cos x) = -\sin x \).
Understanding these basic derivatives:

• For \(\cos x\), the derivative is \(-\sin x \)
• For \(\sin x\), remember the derivative \( \cos x\)

This knowledge allows you to tackle problems involving trigonometric derivatives with confidence. It also prepares you for finding composite derivatives when trigonometric functions are involved, like in the chain rule scenario.

Differentiation

Differentiation is the process of finding the derivative of a function. It measures how a function changes as its input changes. This technique is crucial in both pure and applied mathematics, providing insight into rates of change and slopes of curves.
When you differentiate a function like \(\cos^2 x\), you are calculating how the output \(y\) changes as \(x\) changes. This involves algebraically manipulating the function and applying calculus principles such as the chain rule. The overall goal is to express \(\frac{dy}{dx}\), which represents the rate of change of \(y\) with respect to \(x\).
To differentiate \(y = \cos^2 x\):

• Use the chain rule to manage the composite aspects of the function
• Identify how trigonometric rules apply to the situation (such as \(\sin x\) and \(\cos x\))

Differentiation allows us to understand more than just static values, instead revealing dynamic changes within mathematical contexts.