homework4: problem 2\n(1 point)\nlet ( f(x)=\frac{\tan (x)-3}{sec (x)} ). find the following:\n1. (…

homework4: problem 2\n(1 point)\nlet ( f(x)=\frac{\tan (x)-3}{sec (x)} ). find the following:\n1. ( f^{prime}(x)=cos (x)+sin (x) )\n2. ( f^{prime}(3)=cos (3)+sin (x) )\nnote: you can earn partial credit on this problem.\npreview my answers submit answers\nyou have attempted this problem 1 time.\nyour overall recorded score is ( 0 % ).\nyou have unlimited attempts remaining.

homework4: problem 2\n(1 point)\nlet ( f(x)=\frac{\tan (x)-3}{sec (x)} ). find the following:\n1. ( f^{prime}(x)=cos (x)+sin (x) )\n2. ( f^{prime}(3)=cos (3)+sin (x) )\nnote: you can earn partial credit on this problem.\npreview my answers submit answers\nyou have attempted this problem 1 time.\nyour overall recorded score is ( 0 % ).\nyou have unlimited attempts remaining.

Answer

Explanation:

Step1: Simplify the function

We know that (\tan(x)=\frac{\sin(x)}{\cos(x)}) and (\sec(x)=\frac{1}{\cos(x)}). So (f(x)=\frac{\tan(x)-3}{\sec(x)}=\sin(x)-3\cos(x))

Step2: Differentiate the function

Using the derivative rules ((\sin(x))^\prime=\cos(x)) and ((\cos(x))^\prime =-\sin(x)), for (y = f(x)=\sin(x)-3\cos(x)), then (f^\prime(x)=\cos(x)+ 3\sin(x))

Step3: Evaluate (f^\prime(3))

Substitute (x = 3) into (f^\prime(x)), we get (f^\prime(3)=\cos(3)+3\sin(3))

Answer:

  1. (f^\prime(x)=\cos(x)+3\sin(x))
  2. (f^\prime(3)=\cos(3)+3\sin(3))