Lecture 6 PDF
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Uploaded by FreeLeaningTowerOfPisa6117
2019
Dr. Ahmed kafafy
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Summary
This lecture document covers various calculus concepts including differentiation techniques like implicit differentiation and parametric differentiation. It provides examples and explains the chain rule in different contexts. The topics are suitable for an undergraduate-level mathematics course.
Full Transcript
Suppose that u is a differ. function of x: Example: Find Set then: 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: Find Example: Find ൈ 10/30/2019...
Suppose that u is a differ. function of x: Example: Find Set then: 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: Find Example: Find ൈ 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: Find the derivative of : “ Product rule” “ Chain Rule” Suppose that: ݂ ֜ ݕሺݑሻ & ݂ ֜ ݑሺݔሻ & ݂ ֜ ݔሺݏሻ and ݐ ݂ ֜ ݏthus: 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: Find ݀ݕȀ݀ ݐgiven that: ൈ ൈ 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Suppose that: ݕൌ ݂ሺݑሻ and ݑൌ ሺݔሻ then: By chain Rule: thus: Theorem: the chain Rule 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Differentiating The Trigonometric Functions . . 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: Find d/dx for: Now, the tangent: 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Thus , the tangent: The other Trigs (LTR) Example: Find 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: Find an equation for the tangent to the curve y = cos x at the point ݔൌ ߨȀ͵ ଵ గ ଵ Since ߨȀ͵ ൌ ֜ the point of tangency is ሺ ǡ ሻ ଶ ଷ ଶ ௗ௬ m= ሺ ݔሻ ൌ െ ݔ ௗ௫ ଷ At ݔൌ ߨȀ͵. This gives ݉ ൌ െ ଶ ଵ ଷ గ The tangent equation: ݕെ ൌ െ ሺݔ െ ሻ ଶ ଶ ଷ (LTR) Find 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information The chain Rule applied on the trigs. 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: Find Example: Find 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Parametric differentiation Suppose we want to find dy/dx for a curve that has been defined parametrically. For example: ݔൌ ʹ ݐ ͷ ݕൌ ͵ ݐଶ We can differentiate both of these equations with respect to the parameter t to give: dx dy =2 and = 6t dt dt Using the chain rule: dy dy dt dy dx = × = dx dt dx dt dt 6t = 2 10/30/2019 = 3t Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Parametric differentiation In general, then, to differentiate a pair of parametric equations we can use the chain rule in the form: dy dy dx = dx dt dt ࢊ࢟ Find in terms of t, for the curve defined by the parametric equations ࢊ࢞ x = cos 2t & y = sin t. Differentiating each equation with respect to t gives: dx dy = 2sin2t = cos t dt dt ݀ݕ ݐ ൌ 10/30/2019 ݀ ݔെʹ ʹ ݐ Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Parametric differentiation A curve is defined by the parametric equations ݔൌ ͵ ݐ ʹ ʹ Ƭ ݕൌ ʹͻ ͵ݐ. ࢊ࢟ Find in terms of t and hence find the coordinates of the points where the ࢊ࢞ gradient of the curve is –1. Differentiating with respect to x gives dx = 2(3t + 2)(3) = 6(3t + 2) dt dy = 6t 2 dt dy dy dx 6t 2 = = dx dt dt 6(3t + 2) t2 = 3t + 2 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Implicit Differentiation Suppose: ݕൌ ݂ ֜ ݔis a differentiable function of ݔ, & it is hard to express y explicitly in terms of x, ௗ௬ ǣ֜ ֜ can be obtained by implicit differentiation. ௗ௫ ௗ௬ Example: Find for: ݕൌ ͳ െ ݔଶ ௗ௫ ௗ௬ ௗ ൌ ሺ ͳ െ ݔଶ ሻ ௗ௫ ௗ௫ భ ௗ ଶ మ ൌ ሺͳ െ ݔሻ ௗ௫ భ ଵ ௗ ൌ ሺͳ െ ݔଶ ሻିమ ሺͳ െ ݔଶሻ ଶ ௗ௫ భ ଶ ଶ ିమ ൌ െ ݔሺͳ െ ݔሻ ଶ ି௫ ൌ ଵି௫ మ ݔ ൌെ 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information ݕ ௗ௬ Example: Find for: ௗ௫ Ans. 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information ௗ௬ Example: Express for: ௗ௫ Chain Rule 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: the curve ,, Find the slope of the tangent line to the curve at the point (1, 2) ? Setting x=1 & y=2 we have: ֜ ֜ ସ The slope of the tangent at point (1,2) is: ହ 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Example: use implicit differentiation to express for Differentiate wrt x: Differentiate again wrt x: Thus We have: thus 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Rational Powers we have for +ve integer n. This can be extended to any rational Exponent Ȁݍ: 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Rational Powers If u is a differentiable function of x, then, by the chain rule we have: Example: Find 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information Rational Powers Example: Find 10/30/2019 Math-1, Dr. Ahmed kafafy, MU-Faculty of Computers & Information
