[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-522825-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-522825-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","differentiation-natural-logs-and-exponentials-calculus-worksheet","Differentiation - Natural Logs and Exponentials - Calculus Worksheet","","This calculus worksheet provides a comprehensive set of practice problems focused on differentiating functions involving natural logarithms and exponential terms. The document covers ten distinct problems requiring the application of the chain rule to solve complex derivatives, including nested logarithmic and exponential expressions. Each problem is presented with its step-by-step differentiation solution, demonstrating proper mathematical techniques for handling transcendental functions and variable exponentiation in 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of functions are covered in the practice set?",{"text":72,"@type":64},"The set includes various types of functions such as natural logs of power functions, exponentials with polynomial exponents, nested natural logs, and combinations of trigonometric and exponential functions.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},522825,1791504581,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & 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\nDifferentiate each function with respect to x.  \n1) y = ln x 3 2) y = e 2x3  \n3) y = ln ln 2x4 4) y = ln ln 3x3  \n5) y = cos ln 4x3 6) y = ee3x2  \n7) y = e (4x3 + 5)2 8) y = ln 4x2 ⋅ (−x3 − 4)  \n9) y = ln (− ~~ ~~x433~~ ~~) 5 10) y = ~~ ~~e~~ ~~e~~ ~~54x2~~ ~~3~~ ~~  \nKuta Software-Infinite Calculus Name   \nDifferentiation-Natural Logs and Exponentials Date  Period   \nDifferentiate each function with respect to x.  \n1) y = ln x 3  \ndydx = ~~ ~~1x3~~ ~~ ⋅ 3x2  \n3  \n= ~~ ~~  \nx  \n3) y = ln ln 2x4  \ndy   1   1   \n= ⋅ ⋅ 8x3 dx ln 2x4 2x4  \n4  \n= ~~ ~~  \nx ln 2x4  \n5) y = cos ln 4x3  \ndy 3 1 2  \n~~ ~~ = −sin ln 4x ⋅ ~~ ~~ ⋅ 12x dx 4x3  \n3sin ln 4x3 = −  \nx  \n(4x3 + 5)2  \n7) y = e  \ndydx = e (4x3 + 5)2 ⋅ 2(4x3 + 5) ⋅ 12x2 = 24x2 e (4x3 + 5)2 (4x3 + 5)  \n9) y = ln (− ~~ ~~x433~~ ~~) 5  \ndydx = 5(~~ ~~x4~~ ~~ ⋅ − 16x3 − x~~ ~~3~~ ~~~~ ~~3~~ ~~ ⋅ 3x2 )  \n5(x3 − 12)  \n= 3 (Rules of logarithms used) x(x − 3)  \n2) y = e 2x3  \ndydx = e 2x3 ⋅ 6x2  \n4) y = ln ln 3x3  \ndydx = ~~ ~~ln~~ ~~13x~~ ~~3 ⋅ ~~ ~~31x3~~ ~~ ⋅ 9x2  \n3  \n= ~~ ~~  \nx ln 3x3  \n3x2  \n6) y = ee  \ndydx = ee3x2 e 3x2 ⋅ 6x  \n3x2 2 = 6xee + 3x  \n8) y = ln 4x2 ⋅ (−x3 − 4)  \ndydx = ln 4x2 ⋅ −3x2 + (−x3 − 4) ⋅ ~~ ~~41x2~~ ~~ ⋅ 8x  \n−3x3 ln 4x2 − 2x3 − 8   \n=  \nx  \ne 5x4  \n10) y = ~~ ~~  \ne 4x2 + 3  \ndydx = e5x4 − (4x2 + 3) (20x3 − 8x)  \n= 4xe5x4 − 4x2 − 3 (5x2 − 2) (Rules of exponents used)  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCainodoj4JN8Z0","https://ap.wps.com/l/cbCainodoj4JN8Z0","pdf",28752,"English","# Practice Problems\n## Solutions and Worked Examples","[{\"question\":\"What specific calculus concept is this worksheet primarily designed to practice?\",\"answer\":\"The worksheet is primarily focused on practicing the application of the chain rule to differentiate functions that involve natural logarithms and exponential expressions.\"},{\"question\":\"How are the solutions presented in the document?\",\"answer\":\"Each differentiation problem is accompanied by a step-by-step derivation, showing the application of the chain rule, power rule, and properties of logarithms to arrive at the final derivative expression.\"},{\"question\":\"What types of functions are covered in the practice set?\",\"answer\":\"The set includes various types of functions such as natural logs of power functions, exponentials with polynomial exponents, nested natural logs, and combinations of trigonometric and exponential functions.\"}]","Differentiation - Natural Logs and Exponentials - Calculus Worksheet | PDF",1791426209]