test","extracted_subject":"MATH","extracted_chapter":"DIFFRENTIATION AND APPLICATIONS OF DERIVATIVES","extracted_topic":"DIFFERENTIATION OF IMPLICIT FUNCTION, PARAMETRIC AND COMPOSITE FUNCTIONS","questionPreviewUrl":"https://cdn.pureessence.tech/question-previews/packed_canvases/canvas_1626.png?top_left_x=1064&top_left_y=1592&width=532&height=246&app=solveeit"},"seo_metadata":{"keywords":[]},"doubt_solver_data":{"concept_summary":[]},"embeddings":[],"is_hidden":false,"_id":"66d23e6f425d76b7ab31c375","slug":"if-fx-2supxsup-and-gx-3supxsup-then-fracf0---g01-f","content":"If f(x) = 2^x and g(x) = 3^x, then \$\\frac{f'(0) - g'(0)}{1 + f'(0)g'(0)}\$ =","options":[{"_id":"68a3977d7f15af4716bac769","text":"\$\\frac{\\log\\frac{2}{3}}{1 + \\log 6}\$","sequence":0,"isCorrect":false},{"_id":"68a3977d7f15af4716bac76a","text":"\$\\frac{\\log 6}{1 + \\log 6}\$","sequence":1,"isCorrect":false},{"_id":"68a3977d7f15af4716bac76b","text":"\$\\frac{\\log\\frac{2}{3}}{1 + \\log 2\\log 3}\$","sequence":2,"isCorrect":true},{"_id":"68a3977d7f15af4716bac76c","text":"0","sequence":3,"isCorrect":false}],"correct_answer":"$$\\frac{\\log\\frac{2}{3}}{1 + \\log 2\\log 3}$","explanation":"f(x) = 2^x\n\n⇒ f ′(x) = 2^x log 2\n\n⇒ f ′(0) = log 2\n\ng(x) = 3^x\n\n⇒ g ′(x) = 3^x log 3\n\n⇒ g ′(0) = log 3\n\n= (log 2 – log 3) / (1 + log 2 log 3)","question_type":"single_choice","appeared_in":[],"created_at":"2024-08-30T21:49:35.457Z","__v":0,"vector_store_id":"7332d9fb-40f7-4de1-b609-4291abba29c8","updated_at":"2024-08-30T21:49:35.457Z","isStarred":false},"params":{"questionSlug":"if-fx-2supxsup-and-gx-3supxsup-then-fracf0---g01-f"}}]}]]

Question: If f(x) = 2<sup>x</sup> and g(x) = 3<sup>x</sup>, then \(\frac{f'(0) - g'(0)}{1 + f'(0)g'(0)}\) =...

Solution