[ { "message_id": "assistant-1762103842320", "query": "Find the derivative of the function f(x)=x^3sin(x^2) and simplify your answer.", "agent_response": { "answer": "```json{ \"overview\": \"To find the derivative of the given function, we apply the product rule and the chain rule.\", \"steps\": [ { \"title\": \"Step 1: Identify the Rules for Differentiation\", \"content\": \"The function $f(x) = x^3 \\sin(x^2)$ is a product of two functions, $u(x) = x^3$ and $v(x) = \\sin(x^2)$. Therefore, we will use the product rule for differentiation, which states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.\\n\\nAdditionally, to find the derivative of $v(x) = \\sin(x^2)$, we will need to apply the chain rule, which states that if $y = g(h(x))$, then $y' = g'(h(x))h'(x)$.\", \"expression\": \"f'(x) = u'(x)v(x) + u(x)v'(x)\" }, { \"title\": \"Step 2: Differentiate Each Part\", \"content\": \"First, let's find the derivative of $u(x) = x^3$:\\n$u'(x) = \\\\frac{d}{dx}(x^3) = 3x^2$\\n\\nNext, let's find the derivative of $v(x) = \\\\sin(x^2)$ using the chain rule. Let $w = x^2$, so $v(x) = \\\\sin(w)$.\\n$\\\\frac{dv}{dw} = \\\\cos(w)$\\n$\\\\frac{dw}{dx} = \\\\frac{d}{dx}(x^2) = 2x$\\n\\nApplying the chain rule, $v'(x) = \\\\frac{dv}{dw} \\\\cdot \\\\frac{dw}{dx} = \\\\cos(x^2) \\\\cdot 2x = 2x \\\\cos(x^2)$.\", \"expression\": \"u'(x) = 3x^2 \\\\quad \\\\text{and} \\\\quad v'(x) = 2x \\\\cos(x^2)\" }, { \"title\": \"Step 3: Apply the Product Rule\", \"content\": \"Now, we substitute $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$ into the product rule formula:\\n$f'(x) = u'(x)v(x) + u(x)v'(x)$\\n$f'(x) = (3x^2)(\\\\sin(x^2)) + (x^3)(2x \\\\cos(x^2))$\\n$f'(x) = 3x^2 \\\\sin(x^2) + 2x^4 \\\\cos(x^2)$\", \"expression\": \"f'(x) = 3x^2 \\\\sin(x^2) + 2x^4 \\\\cos(x^2)\" }, { \"title\": \"Step 4: Simplify the Answer\", \"content\": \"We can simplify the expression by factoring out the common term $x^2$ from both terms:\\n$f'(x) = x^2 (3 \\\\sin(x^2) + 2x^2 \\\\cos(x^2))$\", \"expression\": \"f'(x) = x^2 (3 \\\\sin(x^2) + 2x^2 \\\\cos(x^2))\" } ], \"final_answer\": \"$f'(x) = x^2 (3 \\\\sin(x^2) + 2x^2 \\\\cos(x^2))$\"}```", "steps": [ { "title": "Solution", "content": "```json\n{\n \"overview\": \"To find the derivative of the given function, we apply the product rule and the chain rule.\",\n \"steps\": [\n {\n \"title\": \"Step 1: Identify the Rules for Differentiation\",\n \"content\": \"The function $f(x) = x^3 \\sin(x^2)$ is a product of two functions, $u(x) = x^3$ and $v(x) = \\sin(x^2)$. Therefore, we will use the product rule for differentiation, which states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.\\n\\nAdditionally, to find the derivative of $v(x) = \\sin(x^2)$, we will need to apply the chain rule, which states that if $y = g(h(x))$, then $y' = g'(h(x))h'(x)$.\",\n \"expression\": \"f'(x) = u'(x)v(x) + u(x)v'(x)\"\n },\n {\n \"title\": \"Step 2: Differentiate Each Part\",\n \"content\": \"First, let's find the derivative of $u(x) = x^3$:\\n$u'(x) = \\\\frac{d}{dx}(x^3) = 3x^2$\\n\\nNext, let's find the derivative of $v(x) = \\\\sin(x^2)$ using the chain rule. Let $w = x^2$, so $v(x) = \\\\sin(w)$.\\n$\\\\frac{dv}{dw} = \\\\cos(w)$\\n$\\\\frac{dw}{dx} = \\\\frac{d}{dx}(x^2) = 2x$\\n\\nApplying the chain rule, $v'(x) = \\\\frac{dv}{dw} \\\\cdot \\\\frac{dw}{dx} = \\\\cos(x^2) \\\\cdot 2x = 2x \\\\cos(x^2)$.\",\n \"expression\": \"u'(x) = 3x^2 \\\\quad \\\\text{and} \\\\quad v'(x) = 2x \\\\cos(x^2)\"\n },\n {\n \"title\": \"Step 3: Apply the Product Rule\",\n \"content\": \"Now, we substitute $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$ into the product rule formula:\\n$f'(x) = u'(x)v(x) + u(x)v'(x)$\\n$f'(x) = (3x^2)(\\\\sin(x^2)) + (x^3)(2x \\\\cos(x^2))$\\n$f'(x) = 3x^2 \\\\sin(x^2) + 2x^4 \\\\cos(x^2)$\",\n \"expression\": \"f'(x) = 3x^2 \\\\sin(x^2) + 2x^4 \\\\cos(x^2)\"\n },\n {\n \"title\": \"Step 4: Simplify the Answer\",\n \"content\": \"We can simplify the expression by factoring out the common term $x^2$ from both terms:\\n$f'(x) = x^2 (3 \\\\sin(x^2) + 2x^2 \\\\cos(x^2))$\",\n \"expression\": \"f'(x) = x^2 (3 \\\\sin(x^2) + 2x^2 \\\\cos(x^2))\"\n }\n ],\n \"final_answer\": \"$f'(x) = x^2 (3 \\\\sin(x^2) + 2x^2 \\\\cos(x^2))$\"\n}\n```", "expression": null } ], "retrieved_from_kb": false, "knowledge_hits": [ { "document_id": "0", "question": "Find the derivative of f(x)=x^3 sin(x) - MyTutor", "answer": "The derivative of f(x)=x^3 sin(x) is x^2(3 sin(x) + x cos(x)), using the product rule.", "similarity": 0.79119295 }, { "document_id": "1", "question": "Solved The derivative of f(x) can be expressed in the form", "answer": "The derivative of f(x) can be expressed in the form f'(x)=g(x)(3sinx+2)2, where g(x) is some function of x. Find g(x).Answer: g(x)=(b", "similarity": 0.78512776 }, { "document_id": "2", "question": "What is the first derivative of the function f(x) =x^3sinx?", "answer": "The first derivative is: u'v+uv'= 3x^2 sin x + x^3 cos x, or, x^2 (3 sin x + x cos x). However, if you mean x^(3 sin x),", "similarity": 0.7173491 }, { "document_id": "3", "question": "The derivative of the function f is f'(x)=x-3sin x^2. Which ...", "answer": "The derivative of the function f is f'(x)=x-3sin x^2. Which interval contains the greatest number of relative minimums of f? A. (-2,2) B. (-3,1) C. (-4", "similarity": 0.6929958 }, { "document_id": "4", "question": "Larson Calculus 2.3 #44: Differentiate f(x) = sin(x)/x^3 ...", "answer": "Let me show you how I find the derivative with the quotient rule.", "similarity": 0.63452125 } ], "citations": [ { "title": "Find the derivative of f(x)=x^3 sin(x) - MyTutor", "url": "https://www.mytutor.co.uk/answers/17399/A-Level/Maths/Find-the-derivative-of-f-x-x-3-sin-x/" }, { "title": "Solved The derivative of f(x) can be expressed in the form", "url": "https://www.chegg.com/homework-help/questions-and-answers/derivative-f-x-expressed-form-f-x-g-x-3sinx-2-2-g-x-function-x-find-g-x--answer-g-x-b-let--q147428016" }, { "title": "What is the first derivative of the function f(x) =x^3sinx?", "url": "https://www.quora.com/What-is-the-first-derivative-of-the-function-f-x-x-3sinx" }, { "title": "The derivative of the function f is f'(x)=x-3sin x^2. Which ...", "url": "https://www.gauthmath.com/solution/1813181098343573/116-Multiple-Choice-The-derivative-of-the-function-is-f-x-x-3sin-x2-Which-interv" }, { "title": "Larson Calculus 2.3 #44: Differentiate f(x) = sin(x)/x^3 ...", "url": "https://www.youtube.com/watch?v=9qbaO_Jihzg" } ], "source": "tavily-sdk", "feedback_required": true, "gateway_trace": [ "input_guardrails_pass", "router.inspect", "router->search", "search.source=tavily-sdk", "output_guardrails_pass" ] }, "feedback": { "thumbs_up": true, "primary_issue": null, "has_better_solution": false, "solution_type": null, "better_solution_text": null, "better_solution_pdf_base64": null, "better_solution_image_base64": null } }, { "message_id": "assistant-1762145260594", "query": "Find the derivative of the function: f(x)=3x^4−5x^2+2squareroot(x)−7/x^3", "agent_response": { "answer": "Thus, the function can be rewritten as:", "steps": [ { "title": "Overview", "content": "Let's find the derivative of the function $f(x)$ step by step.", "expression": null } ], "retrieved_from_kb": false, "knowledge_hits": [ { "document_id": "0", "question": "Derivative Calculator - Mathway", "answer": "Upgrade Calculators Help Sign In Sign Up # Derivative Calculator **Step 1:** Enter the function you want to find the derivative of in the editor. > The Derivative Calculator supports solving first, second...., fourth derivatives, as well as implicit differentiation and finding the zeros/roots. You can also get a better visual and understanding of the function by using our graphing tool. > Chain Rule: ddx[f(g(x))]=f'(g(x))g'(x) **Step 2:** Click the blue arrow to submit. Choose **\"Find the Derivative\"** from the topic selector and click to see the result! (2x−9)10 using Amazon.Auth.AccessControlPolicy; Mathway requires javascript and a modern browser. Please ensure that your password is at least 8 characters and contains each of the following: * a special character: @$#!%\\*?&", "similarity": 0.98579 }, { "document_id": "1", "question": "Derivative Calculator - Math Portal", "answer": "* Calculators Math Calculators, Lessons and Formulas * Calculators * Calculators * Derivative Calculator # Derivative calculator * Fraction Calculator * Percentage Calculator Polynomials * Factoring Polynomials * Simplify Polynomials * Multiplication / Division Solving Equations * Polynomial Equations * Square Calculator * Rectangle Calculator * Circle Calculator * Hexagon Calculator * Rhombus Calculator * Trapezoid Calculator * Distance calculator * Midpoint Calculator * Triangle Calculator * Division * Determinant Calculator Calculus Calculators * Limit Calculator * Derivative Calculator * Integral Calculator Equations * Probability Calculator * T-Test Calculator Financial Calculators * Amortization Calculator * Annuity Calculator Other Calculators Related calculators Limit calculator Integral calculator Derivative calculator – Widget Code **Widget preview:** Derivative calculator Online Math Calculators", "similarity": 0.98278 }, { "document_id": "2", "question": "Derivative Calculator - Symbolab", "answer": "*Example* : Find the derivative of $f\\left(x\\right)=\\frac{1}{x}$. scientific calculator inverse calculator simplify calculator distance calculator fractions calculator interval notation calculator cross product calculator probability calculator derivative calculator series calculator ratios calculator statistics calculator integral calculator inverse laplace transform calculator rounding calculator gcf calculator algebra calculator tangent line calculator trigonometry calculator log calculator standard deviation calculator linear equation calculator antiderivative calculator laplace transform calculator quadratic equation calculator domain calculator decimals calculator limit calculator equation solver definite integral calculator matrix inverse calculator matrix calculator system of equations calculator calculus calculator slope calculator long division calculator factors calculator polynomial calculator square root calculator implicit differentiation calculator word problem solver differential equation calculator average calculator synthetic division calculator", "similarity": 0.97997 }, { "document_id": "3", "question": "Calculus - How to find the derivative of a function using the power rule", "answer": "Calculus - How to find the derivative of a function using the power rule\nMySecretMathTutor\n241000 subscribers\n6381 likes\n582542 views\n5 Sep 2013\nThis video shows how to find the derivative of a function using the power rule. Remember that this rule only works on functions of the form x^n where n is not equal to zero. For more videos please visit http://www.mysecretmathtutor.com\n189 comments\n", "similarity": 0.97435 }, { "document_id": "4", "question": "Finding the Derivative of a Square Root Function Using Definition of ...", "answer": "Finding the Derivative of a Square Root Function Using Definition of a Derivative\nPatrick J\n1400000 subscribers\n1825 likes\n268324 views\n11 Dec 2011\nIn this video we compute the derivative of a square root function, namely f(x) = sqrt (2x + 1) by using the definition of a derivative. We could use the power rule along with the chain rule as well of course, but this what one is often asked in the beginning of a calculus course.\nNote that to help simplify the expression for the derivative, we will make use of the conjugate, a common technique to help simplify these types of expressions with square roots.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n#math, #algebra, #khanacademy, #khan, #geometry, #education, #montessori, #calculus, #homeschooling, #functions, #notation, #organicchemistry, #organicchemistrytutor, #calculus, #derivative, #powerrule, #limits, #limit, #graph, #graphing, #conjugate, #infinity, #patrickjmt, #rational, #shortcut, #trick, #commondenominator, #fraction, #university, #admissions, #test #continuity #onesided #infinite #absolutevalue #tangents #tangentsandnormals #tangentline #definition #hindi #india\n126 comments\n", "similarity": 0.96945 } ], "citations": [ { "title": "Derivative Calculator - Mathway", "url": "https://www.mathway.com/Calculator/derivative-calculator" }, { "title": "Derivative Calculator - Math Portal", "url": "https://www.mathportal.org/calculators/calculus/derivative-calculator.php" }, { "title": "Derivative Calculator - Symbolab", "url": "https://www.symbolab.com/solver/derivative-calculator" }, { "title": "Calculus - How to find the derivative of a function using the power rule", "url": "https://www.youtube.com/watch?v=pBc4Udqw330" }, { "title": "Finding the Derivative of a Square Root Function Using Definition of ...", "url": "https://www.youtube.com/watch?v=3bmw9UQxbRI" } ], "source": "tavily-sdk", "feedback_required": true, "gateway_trace": [ "input_guardrails_pass", "router.inspect", "router->search", "search.source=tavily-sdk", "output_guardrails_pass" ] }, "feedback": { "thumbs_up": false, "primary_issue": "unclear", "has_better_solution": false, "solution_type": null, "better_solution_text": null, "better_solution_pdf_base64": null, "better_solution_image_base64": null } } ]