To determine
To find: The derivative of the function f(x)=secx−x.
Answer
The derivative of f(x) is f′(x)=secxtanx−1_.
Explanation
Given:
The function is f(x)=secx−x.
Derivative rules:
(1) Difference Rule: ddx[f(x)−g(x)]=ddx[f(x)]−ddx[g(x)]
(2) Power Rule: ddx(xn)=nxn−1
Calculation:
Obtain the derivative of f(x).
f′(x)=ddx(f(x)) =ddx(secx−x)
Apply the difference rule (1),
f′(x)=ddx[secx]−ddx[x]=secxtanx−ddx[x]
Apply the power rule (2) and simplify further,
f′(x)=secxtanx−[1x1−1]=secxtanx−1
Therefore, the derivative of f(x) is f′(x)=secxtanx−1_.
To determine
To check: The derivatives of f(x) obtained in part (a) is reasonable or not by using the graphs of f(x) and f′(x).
Answer
The derivatives of f(x) is reasonable.
Explanation
Graph:
Use the online graphing calculator and draw the graph of f(x)=secx−x and f′(x)=secxtanx−1 as shown below in Figure 1.

Observation:
From Figure 1, it is noticed that,
If f′(x) is positive then f(x) is increasing function.
If f′(x) is negative then f(x) is decreasing function.
If f′(x) crosses the x axis (f′(x)=0), then f(x) is local extrema (that is, local minimum or local maximum).
Therefore, it can be concluded that the derivative of the function f(x)=secx−x is reasonable.