Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License We also see that where f ( x ) = sin x f ( x ) = sin x is increasing, f ′ ( x ) = cos x > 0 f ′ ( x ) = cos x > 0 and where f ( x ) = sin x f ( x ) = sin x is decreasing, f ′ ( x ) = cos x < 0. Notice that at the points where f ( x ) = sin x f ( x ) = sin x has a horizontal tangent, its derivative f ′ ( x ) = cos x f ′ ( x ) = cos x takes on the value zero. = cos x Simplify.įigure 3.27 shows the relationship between the graph of f ( x ) = sin x f ( x ) = sin x and its derivative f ′ ( x ) = cos x. = lim h → 0 ( sin x ( cos h − 1 h ) + cos x ( sin h h ) ) Factor out sin x and cos x. = lim h → 0 ( sin x cos h − sin x h + cos x sin h h ) Regroup. = lim h → 0 sin x cos h + cos x sin h − sin x h Use trig identity for the sine of the sum of two angles. d d x sin x = lim h → 0 sin ( x + h ) − sin x h Apply the definition of the derivative. Recall that for a function f ( x ), f ( x ) ,ĭ d x sin x = lim h → 0 sin ( x + h ) − sin x h Apply the definition of the derivative. We begin our exploration of the derivative for the sine function by using the formula to make a reasonable guess at its derivative. Derivatives of the Sine and Cosine Functions Being able to calculate the derivatives of the sine and cosine functions will enable us to find the velocity and acceleration of simple harmonic motion. We begin with the derivatives of the sine and cosine functions and then use them to obtain formulas for the derivatives of the remaining four trigonometric functions. In this section we expand our knowledge of derivative formulas to include derivatives of these and other trigonometric functions. Simple harmonic motion can be described by using either sine or cosine functions. One of the most important types of motion in physics is simple harmonic motion, which is associated with such systems as an object with mass oscillating on a spring. 3.5.3 Calculate the higher-order derivatives of the sine and cosine. 3.5.2 Find the derivatives of the standard trigonometric functions. Here is a set of assignement problems (for use by instructors) to accompany the Chain Rule section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University.3.5.1 Find the derivatives of the sine and cosine function.
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