let $f(x)=7(\\sin(x))^{x}$. find $f(1)$.\n$f(1)=$
let $f(x)=7(\\sin(x))^{x}$. find $f(1)$.\n$f(1)=$
Answer
Explanation:
Step1: Apply the formula (y = a^{x}\Rightarrow y'=a^{x}\ln a) and product rule
Given (f(x) = 7(\sin(x))^{x}). Let (y = (\sin(x))^{x}), then (\ln y=x\ln(\sin x)). Differentiate both sides with respect to (x): (\frac{y'}{y}=\ln(\sin x)+x\cdot\frac{\cos x}{\sin x}) So (y' = (\sin x)^{x}\left(\ln(\sin x)+x\cot x\right)) Then (f'(x)=7(\sin x)^{x}\left(\ln(\sin x)+x\cot x\right))
Step2: Substitute (x = 1)
When (x = 1), (\sin(1)) remains (\sin(1)), (\cot(1)=\frac{\cos(1)}{\sin(1)}) (f'(1)=7(\sin(1))^{1}\left(\ln(\sin(1)) + 1\times\cot(1)\right)) (=7\sin(1)\left(\ln(\sin(1))+\cot(1)\right))
Answer:
(7\sin(1)\left(\ln(\sin(1))+\cot(1)\right))