if ( f(x)=\frac{\tan x - 3}{sec x} )( f(x)=)( f(1)=)

Answer:

$f^{\prime}(x)=\cos x + 3\tan x\sec x$; $f^{\prime}(1)=\cos 1 + 3\tan 1\sec 1$

Explanation:

Step1: Rewrite 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 sum - difference rule of differentiation $(u\pm v)^\prime=u^\prime\pm v^\prime$ and the basic differentiation formulas $(\sin x)^\prime=\cos x$ and $(\cos x)^\prime=-\sin x$. For $y = \sin x-3\cos x$, then $y^\prime=(\sin x)^\prime-(3\cos x)^\prime$. Since $(k\cdot u(x))^\prime=k\cdot u^\prime(x)$ (where $k = 3$ and $u(x)=\cos x$), we have $y^\prime=\cos x+3\sin x\cdot\frac{1}{\cos x}=\cos x + 3\tan x\sec x$

Step3: Evaluate the derivative at $x = 1$

Substitute $x = 1$ into $f^{\prime}(x)$. We get $f^{\prime}(1)=\cos(1)+3\tan(1)\sec(1)$ (where the angles are in radians)