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Chapter 3: Problem 25

Find \(d y / d x\) for the following functions. $$y=\sin x \cos x$$

Short Answer

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

Step by step solution

01

As we have the equation \(y=\sin x \cos x\), the functions involved are \(\sin x\) and \(\cos x\). Let's denote them as \(u(x)=\sin x\) and \(v(x)=\cos x\). #Step 2: Find the derivatives of u and v with respect to x#

Now, we'll find the derivatives of \(u(x)\) and \(v(x)\) with respect to x. We know that the derivative of \(\sin x\) is \(\cos x\), and the derivative of \(\cos x\) is \(-\sin x\). Therefore, $$u'(x)=\frac{d(\sin x)}{dx}=\cos x$$ and $$v'(x)=\frac{d(\cos x)}{dx}=-\sin x$$ #Step 3: Apply the product rule to find the derivative of y#
02

According to the product rule, if \(y = uv\), then: $$\frac{dy}{dx} = u'v + uv'$$ So, substituting our earlier results for \(y\), \(u'\), and \(v'\), we get: $$\frac{dy}{dx} = (\cos x)(\cos x)+(\sin x)(-\sin x)$$ #Step 4: Simplify the expression for dy/dx#

Let's now simplify the expression we obtained in the previous step. This gives us: $$\frac{dy}{dx} = \cos^2 x - \sin^2 x$$ Solution: Our final answer for the derivative of the given function is: $$\frac{dy}{dx} = \cos^2 x - \sin^2 x$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Product Rule
Understanding the product rule is essential for differentiating expressions where two or more functions are multiplied together. It states that if you have two functions, let's call them u(x) and v(x), the derivative of their product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x), where u'(x) and v'(x) are the derivatives of u(x) and v(x) respectively.

This becomes particularly useful in trigonometry where products of sine and cosine functions are common. For example, when differentiating y = sin(x)cos(x), we apply the product rule to find dy/dx. In our exercise, we first differentiated sin(x) and cos(x) individually and then applied the product rule to get the derivative of their product. The result is a combination of the derivatives, as seen in the step by step solution.
Differentiation
Differentiation is a fundamental concept in calculus, referring to the process of finding the derivative of a function, which represents the function's rate of change. The derivative of a function at a point is the slope of the tangent line to the function's graph at that point. It tells us how much the function 'y' changes for a small change in 'x'.

In the context of trigonometric functions, differentiation helps us understand the behavior of waves and oscillations since these functions model periodic phenomena. Sine and cosine functions, for example, have well-defined derivatives that are crucial for solving real-world problems in physics and engineering.
Trigonometry
Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field has expanded, and now trigonometric functions like sine, cosine, and tangent are used to describe periodic phenomena.

For instance, these functions are used to model sound waves, light waves, and the motion of pendulums. When it comes to calculus, trigonometric functions are often differentiated or integrated, allowing us to solve problems involving motion and change.
Chain Rule
The chain rule is another crucial differentiation technique used when dealing with composite functions - functions made up by combining other functions. The chain rule states that to differentiate a composite function, you differentiate the outer function, then multiply it by the derivative of the inner function.

Although it was not directly used in our exercise, many trigonometric functions involve composition, such as sin(g(x)) where g(x) is some other function. Knowing when to apply the chain rule, and how to do it correctly, is essential for accurately finding derivatives of complex functions.

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Most popular questions from this chapter

Horizontal tangents The graph of \(y=\cos x \cdot \ln \cos ^{2} x\) has seven horizontal tangent lines on the interval \([0,2 \pi] .\) Find the approximate \(x\) -coordinates of all points at which these tangent lines occur.

The bottom of a large theater screen is \(3 \mathrm{ft}\) above your eye level and the top of the screen is \(10 \mathrm{ft}\) above your eye level. Assume you walk away from the screen (perpendicular to the screen) at a rate of \(3 \mathrm{ft} / \mathrm{s}\) while looking at the screen. What is the rate of change of the viewing angle \(\theta\) when you are \(30 \mathrm{ft}\) from the wall on which the screen hangs, assuming the floor is horizontal (see figure)?

Jean and Juan run a one-lap race on a circular track. Their angular positions on the track during the race are given by the functions \(\theta(t)\) and \(\varphi(t),\) respectively, where \(0 \leq t \leq 4\) and \(t\) is measured in minutes (see figure). These angles are measured in radians, where \(\theta=\varphi=0\) represent the starting position and \(\theta=\varphi=2 \pi\) represent the finish position. The angular velocities of the runners are \(\theta^{\prime}(t)\) and \(\varphi^{\prime}(t)\). a. Compare in words the angular velocity of the two runners and the progress of the race. b. Which runner has the greater average angular velocity? c. Who wins the race? d. Jean's position is given by \(\theta(t)=\pi t^{2} / 8 .\) What is her angular velocity at \(t=2\) and at what time is her angular velocity the greatest? e. Juan's position is given by \(\varphi(t)=\pi t(8-t) / 8 .\) What is his angular velocity at \(t=2\) and at what time is his angular velocity the greatest?

Proof by induction: derivative of \(e^{k x}\) for positive integers \(k\) Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually \(k=1\) ). In the second step, the statement is assumed to be true for \(k=n\), and the statement is proved for \(k=n+1,\) which concludes the proof. a. Show that \(\frac{d}{d x}\left(e^{k x}\right)=k e^{k x},\) for \(k=1\) b. Assume the rule is true for \(k=n\) (that is, assume \(\left.\frac{d}{d x}\left(e^{n x}\right)=n e^{n x}\right),\) and show this assumption implies that the rule is true for \(k=n+1\). (Hint: Write \(e^{(n+1) x}\) as the product of two functions and use the Product Rule.)

a. Use derivatives to show that \(\tan ^{-1} \frac{2}{n^{2}}\) and \(\tan ^{-1}(n+1)-\tan ^{-1}(n-1)\) differ by a constant. b. Prove that \(\tan ^{-1} \frac{2}{n^{2}}=\tan ^{-1}(n+1)-\tan ^{-1}(n-1)\)
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