Higher Order Derivatives (Part 1)

In this video, we continue our study of Higher Order Derivatives in Calculus, focusing on how to differentiate functions repeatedly to obtain second and higher derivatives in a clear, step-by-step manner. We solve a detailed example involving a rational trigonometric function and carefully apply the quotient rule alongside trigonometric identities to simplify the result. We show how to: Apply the quotient rule correctly Differentiate trigonometric functions step by step Simplify using identities such as sin^2(x)+cos^2(x)=1 Obtain clean final answers for first and second derivatives TIMESTAMPS ⏱️ 0:00-0:40 Introduction to higher derivatives Meaning of first, second, and higher derivatives Notation: y′,,y′′,y′′′ ⏱️ 0:40 – 2:10 First example: y=cos^2(x) Rewriting as cosx⋅cosx Applying product rule Finding the first derivative y′ ⏱️ 2:10 – 3:10 Second derivative of y=cos^2(x) Differentiating again Simplifying using identities Result leading to −2cos2x ⏱️ 3:10 – 3:30 Transition to second question Introduction of quotient rule problem y=cosx/1−sinx ⏱️ 3:30 – 5:00 First derivative of quotient function Applying quotient rule Simplifying using sin^2(x)+cos^2(x) =1 Final simplification of y′ ⏱️ 5:00 – 5:50 Second derivative + conclusion Differentiating again Final result of second derivative Ending remarks and call to action By the end of this lesson, you will clearly understand how to handle more advanced differentiation problems involving fractions and trigonometric expressions. This video is perfect for students studying MTH 102 (Calculus I) and engineering mathematics courses. 👍 Don’t forget to like, share, and subscribe for more clear and simplified mathematics tutorials. For collaborations, tutoring, or academic support, contact: [Email: [email protected] ] Twitter : @MathsciC49603 #Calculus #HigherOrderDerivatives #Derivatives #MathTutorial #QuotientRule #MTH102 #EngineeringMath #Differentiation #StudyMath #Mathematics