Calculus 1 Lecture 2.5: Finding Derivatives of Trigonometric Functions

Comments · 276

• AP

  @AP-pm9qy · 8 years ago (edited)

  Right now its 3 am on a Saturday and I'm staying up to watch Professor Leonard's calc vids. I told myself just one more and here I am. As someone who spends their time on Netflix and suffers from chronic laziness, Professor Leonard managed to engage me so much so that I'm binge watching his videos and taking notes like its a new and hot Netflix series that came out. You sir are legendary.

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• SA

  @SakibLH44 · 11 years ago

  I wish my tuition money was going to you instead of my half-assing prof at UTA

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• JA

  @jackie-chan4142 · 9 months ago

  Who&apos;s here in 2025?<br>Even till today this guy is saving souls in calculus!!!! Thank you Professor Leonard.<br>I have a bit of a story. So me and my friends placed a bet that I couldn&apos;t learn trig and calculus in time for our math team event next month. I lowkey thought I couldn&apos;t do it until a few weeks ago when I found your videos. &nbsp;You saved my ass trying to teach myself all of this in 2 months, and you actually made it fun!!!!! Lowkey wish I had you as a teacher for all my previous math years I could&apos;ve saved years D:

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• JI

  @jie-wl7iv · 8 months ago

  Who is in 2026!!!!!

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• CA

  @carlofatx · 11 years ago

  You are a very mindful professor. My professor will make anybody feel like they&apos;re stupid for asking any question about algebra.

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• AM

  @ambyjay · 5 years ago

  I can’t believe I just found this guy 6 weeks into the semester 😵<br>Thank you in advance for getting me through calculus!

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• IM

  @imy_mac · 9 months ago

  11 years later this man is still saving lives lol.

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• DA

  @davidsphere43 · 4 years ago

  After all these years, I have found the perfect video series. It&apos;s 6:43 am, I stayed up all night studying for an exam today (probably a bad idea), and just found these videos, and now I can only be jealous that I dont have a professor this good at teaching.<br><br>Hopefully watching these lessons will save me today

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• BR

  @brianwahome2897 · 9 years ago

  I took a weekend break when you said: &quot;We will get to the rest of it next time&quot; at <a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=233">3:53</a>... Excellent work Prof.

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• SI

  @SIMPHIWENdlovu-o7q · 3 years ago

  Even today, 2023. Professor &nbsp;Leonard &nbsp;videos are still relevant &nbsp;and essential. &nbsp;I can spend all day watching &nbsp;his videos. Makes maths look easier. I was not performing really well on my studies &nbsp;but since I started watching him. My maths results are getting better and better. Masterpiece in Maths. FROM SOUTH AFRICA

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• IÑ

  @iñigote · 1 year ago (edited)

  Introduction and Motivation<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=8">0:08</a>]. Introduction to trigonometric functions and their importance in later calculus courses.<br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=57">0:57</a>]. Presentation of two fundamental concepts necessary for the demonstration:<br> &nbsp; &nbsp;○ Trigonometric identity: sin(X) / X = 1.<br> &nbsp; &nbsp;○ Trigonometric identity: [1 - cos(X)] / X = 0.<br><br>Demonstration of the Derivative of the Sine Function<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=127">2:07</a>]. Objective: find the derivative of 𝑓(x) = sin(x), denoted as 𝑓&apos;(x).<br> &nbsp; &nbsp;○ Reminder of the concept of derivative as a limit:<br> &nbsp; &nbsp;○ 𝑓&apos;(x) = lim_(h → 0) [𝑓(x + h) - 𝑓(x)] / h.<br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=175">2:55</a>]. Application of the limit concept to the sine function:<br> &nbsp; &nbsp;○ 𝑓(x) = sin(x).<br> &nbsp; &nbsp;○ 𝑓(x + h) = sin(x + h).<br> &nbsp; &nbsp;○ Restriction: the expression x + h inside the sine function cannot <br> &nbsp; &nbsp; &nbsp; &nbsp;be separated without using trigonometric identities.<br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=212">3:32</a>]. First step of the demonstration:<br> &nbsp; &nbsp;○ 𝑓&apos;(x) = lim_(h → 0) [sin(x + h) - sin(x)] / h.<br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=300">5:00</a>]. Use of the angle sum trigonometric identity for sine:<br> &nbsp; &nbsp;○ sin(x + h) = sin(x) ⋅ cos(h) + cos(x) ⋅ sin(h).<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=372">6:12</a>]. Separation of the expression into two fractions with denominator h.<br> &nbsp; &nbsp;○ Factoring out sin(x) in the first term.<br>●Algebraic manipulation to obtain the known trigonometric limits:<br> &nbsp; &nbsp;○ lim_(h → 0) sin(h) / h = 1.<br> &nbsp; &nbsp;○ lim_(h → 0) [1 - cos(h)] / h = 0.<br>●Application of the limits and simplification to obtain the derivative of sine:<br> &nbsp; &nbsp;○ 𝑓&apos;(x) = cos(x).<br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=675">11:15</a>]. Interpretation: the derivative of the sine function is the cosine function, meaning <br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; that the slope of the sine curve at any point is equal to the value of the cosine at that point.<br><br>Derivatives of Other Trigonometric Functions<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=770">12:50</a>]. Presentation of the derivatives of the six main trigonometric functions:<br> &nbsp; &nbsp;1. d/dx [sin(x)] = cos(x).<br> &nbsp; &nbsp;2. d/dx [cos(x)] = -sin(x).<br> &nbsp; &nbsp;3. d/dx [tan(x)] = sec²(x).<br> &nbsp; &nbsp;4. d/dx [sec(x)] = sec(x)tan(x).<br> &nbsp; &nbsp;5. d/dx [csc(x)] = -csc(x)cot(x).<br> &nbsp; &nbsp;6. d/dx [cot(x)] = -csc²(x).<br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=821">13:41</a>]. The derivatives of tangent, secant, cosecant, and cotangent can be obtained using <br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; the quotient rule and the derivatives of sine and cosine.<br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=977">16:17</a>]. Observation on the relationships between the derivatives of trigonometric functions,<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; which can help in memorizing them.<br><br><br>Examples of Applying Trigonometric Derivatives<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=1148">19:08</a>]. Example 1: Find the derivative of y = x ⋅ sin(x).<br> &nbsp; &nbsp;○ [<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=1173">19:33</a>]. Identification of the need to use the product rule.<br> &nbsp; &nbsp;○ Application of the product rule:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ d/dx [x ⋅ sin(x)] = (d/dx [x] ⋅ sin(x)) + (x ⋅ d/dx [sin(x)]).<br> &nbsp; &nbsp;○ Simplification to obtain the derivative:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ d/dx [x ⋅ sin(x)] = sin(x) + x ⋅ cos(x).<br> &nbsp; &nbsp;○ Interpretation: the obtained expression represents the slope function for <br> &nbsp; &nbsp; &nbsp; &nbsp;the curve y = x sin(x).<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=1295">21:35</a>]. Example 2: Find the equation of the tangent line to the curve y = x ⋅ sin(x) at the point x = π/2.<br> &nbsp; &nbsp;○ Requirements for the equation of a line: a point and the slope.<br> &nbsp; &nbsp;○ Obtaining the slope (m) at x = π/2:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ m = sin(π/2) + (π/2) ⋅ cos(π/2) = 1.<br> &nbsp; &nbsp;○ Obtaining the point (π/2, y) by substituting x = π/2 into the original equation:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ y = (π/2) ⋅ sin(π/2) = π/2.<br> &nbsp; &nbsp;○ Use of the point-slope formula (y - y₁ = m(x - x₁)) to obtain the equation of the tangent line.<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=1580">26:20</a>]. Example 3: Find the derivative of y = sin(x) / (1 + cos(x)).<br> &nbsp; &nbsp;○ Identification of the need to use the quotient rule.<br> &nbsp; &nbsp;○ Application of the quotient rule:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ d/dx [sin(x) / (1 + cos(x))] = {(d/dx [sin(x)] ⋅ (1 + cos(x))) - (sin(x) ⋅ d/dx [1 + cos(x)])} / (1 + cos(x))².<br> &nbsp; &nbsp;○ Calculation of individual derivatives and simplification.<br> &nbsp; &nbsp;○ [<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=1902">31:42</a>]. Use of the trigonometric identity cos²(x) + sin²(x) = 1 to simplify the expression.<br> &nbsp; &nbsp;○ Additional simplification to obtain the derivative:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ d/dx [sin(x) / (1 + cos(x))] = 1 / (1 + cos(x)).<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=2020">33:40</a>]. Example 4: Calculation of higher-order derivatives for the function sin(x).<br> &nbsp; &nbsp;○ Calculation of the second, third, and fourth derivatives of sin(x).<br> &nbsp; &nbsp;○ Observation on the cyclical nature of the derivatives of the sine function.<br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=2145">35:45</a>]. Example 5 (verbal problem): Modeling the motion of a spring with a mass.<br> &nbsp; &nbsp;○ Description of the problem: a spring with a mass is stretched 3 cm from its <br> &nbsp; &nbsp; &nbsp;resting position and released, oscillating without friction.<br> &nbsp; &nbsp;○ [<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=2499">41:39</a>]. Modeling the position of the spring with a cosine function: y(t) = -3⋅cos(t).<br> &nbsp; &nbsp;○ Objective: find the velocity function of the spring.<br> &nbsp; &nbsp;○ [<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=2622">43:42</a>]. Relationship between velocity and position: velocity is the derivative of position.<br> &nbsp; &nbsp;○ [<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=2644">44:04</a>]. Calculation of the derivative of the position function:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ y&apos;(t) = 3 ⋅ sin(t).<br> &nbsp; &nbsp;○ Possible additional questions about the problem:<br> &nbsp; &nbsp; &nbsp; &nbsp;■ Find the points where the velocity is zero.<br> &nbsp; &nbsp; &nbsp; &nbsp;■ Find the equation of the tangent line at a specific point.<br><br><br>●[<a href="https://www.youtube.com/watch?v=RJJSiNz5oto&amp;t=2758">45:58</a>]. Introduction to the chain rule.<br> &nbsp; &nbsp;○ Presentation of examples of derivatives that are difficult to calculate *without the chain rule*.<br> &nbsp; &nbsp;○ Description of the utility of the chain rule to simplify the calculation of derivatives<br> &nbsp; &nbsp; &nbsp; of composite functions.

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• N3

  @n3bulou5 · 3 years ago

  This is fascinating after taking your Precalculus/Trigonometry course. Knowing how you can use sin and cos to model oscillations and now being able to obtain the rate of change at a given time is really enlightening. I tried going through this course several years ago, and this lecture was so perplexing at the time, not knowing the prerequisites, that I gave up. Now 4 years later it seems really easy. Filling in that missing puzzle piece in your course list was clutch sir, and needed for so many of us.

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