let us assume that for some function $f$, we have $f(0) = 6$ and $f(0) = 9$. let $f(x) = f(x)\tan x$. find $f(0)$.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Product Rule Differentiation", "Trigonometric Derivatives" ], "new_concepts": [], "current_concepts": [ "Product Rule Differentiation", "Trigonometric Derivatives" ] } ## Apply the product rule $$ F'(x) = f'(x)\tan x + f(x)\sec^2 x $$ ## Substitute the given values at x = 0 $$ F'(0) = f'(0)\tan(0) + f(0)\sec^2(0) $$ ## Evaluate the trigonometric and given values $$ \begin{aligned} \tan(0) &= 0 \ \sec(0) &= 1 \implies \sec^2(0) = 1 \ F'(0) &= 9 \cdot 0 + 6 \cdot 1 = 6 \end{aligned} $$ Let us assume that for some function $f$, we have $f(0) = 6$ and $f'(0) = 9$. Let $F(x) = f(x)\tan x$. Find $F'(0)$. $F'(0) =$ 6 { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Product Rule Differentiation" ] }

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Explanation:

Apply the product rule

Substitute the given values at x = 0

Evaluate the trigonometric and given values

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