Answer: (sinx) tanx × [sec²x × ln(sinx) + 1].
- A (sinx)<sup>tanx</sup> × [sec²x × ln(sinx) + 1]
- B (sinx)<sup>tanx</sup> × sec²x, omitting the logarithmic term
- C (sinx)<sup>tanx</sup> × ln(sinx), omitting the secant-squared term
- D tanx × (sinx)<sup>tanx-1</sup>, treating it like a simple power rule
Correct answer: A. (sinx)<sup>tanx</sup> × [sec²x × ln(sinx) + 1]
Explanation: Take ln: y = tanx × ln(sinx). Differentiate: y'/y = sec²x × ln(sinx) + tanx × cosx/sinx = sec²x lnsinx + 1. y' = y × [sec²x lnsinx + 1].
As point Q slides along the curve toward P, the secant line PQ rotates into the tangent line at P, whose slope is the derivative.
Concept context
Rate of change, differentiation rules, and applications