Find the Derivative of #sec x# using first principle?

Question

Samantha Adkison · Accepted Answer

# d/dx sec x =tanx secx #

Define the function: # f(x)=secx # Using the limit definition of the derivative, we have: # f'(x) = lim_(h rarr 0) (f(x+h)-f(x))/h # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (sec(x+h)-sec(x))/h # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (1/cos(x+h)-1/cos(x))/h # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) ((cosx-cos(x+h))/(cos(x+h)cos(x)))/h # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (cosx-cos(x+h))/(hcos(x+h)cos(x)) # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (cosx-(cosxcos h-sin x sin h))/(hcos(x+h)cos(x)) # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (cosx-cosxcos h+sin x sin h)/(hcos(x+h)cos(x)) # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (cosx(1-cos h)+sin x sin h)/(hcos(x+h)cos(x)) # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (cosx(1-cos h))/(hcos(x+h)cos(x))+(sin x sin h)/(hcos(x+h)cos(x)) # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) ((1-cos h))/(hcos(x+h))+(tan x sin h)/(hcos(x+h)) # # \ \ \ \ \ \ \ \ = lim_(h rarr 0) (1-cos h)/h * sec(x+h)+(sin h)/h * tanx sec(x+h) # Then we use two standard calculus limits: # lim_(theta rarr 0) (1-cos theta)/theta = 0 # and # lim_(theta rarr 0) (sin theta)/theta = 1# Which gives us: # f'(x) = 0 * sec(x) + 1 * tanx sec(x) # # \ \ \ \ \ \ \ \ = tanx secx #

Ezekiel Alexander · Answer

undefined To find the derivative of sec(x) using the first principles, we start with the definition of the secant function:

sec(x) = 1 / cos(x)

Now, let f(x) = sec(x), and we want to find f'(x) using the definition of the derivative:

f'(x) = lim(h->0) [sec(x + h) - sec(x)] / h

= lim(h->0) [1 / cos(x + h) - 1 / cos(x)] / h

= lim(h->0) [(cos(x) - cos(x + h)) / (cos(x) * cos(x + h))] / h

= lim(h->0) [(cos(x) - (cos(x)cos(h) - sin(x)sin(h))) / (cos(x) * cos(x + h))] / h

= lim(h->0) [(cos(x) - cos(x)cos(h) + sin(x)sin(h)) / (cos(x) * cos(x + h))] / h

= lim(h->0) [(cos(x)(1 - cos(h)) + sin(x)sin(h)) / (cos(x) * cos(x + h))] / h

= lim(h->0) [(cos(x)(1 - cos(h)) + sin(x)sin(h)) / (cos(x) * cos(x) * (1 - h^2/2) + sin(x)sin(x) * (h - h^3/6) + O(h^4))] / h

= lim(h->0) [(cos(x)(1 - cos(h)) + sin(x)sin(h)) / (cos^2(x) * (1 - h^2/2) + sin^2(x) * (h - h^3/6) + O(h^4))] / h

= lim(h->0) [(cos(x)(1 - cos(h)) + sin(x)sin(h)) / (cos^2(x) - cos^2(x) * h^2/2 + sin^2(x) * h - sin^2(x) * h^3/6 + O(h^4))] / h

= lim(h->0) [(cos(x)(1 - cos(h)) + sin(x)sin(h)) / (cos^2(x) + sin^2(x) - (cos^2(x) * h^2/2 - sin^2(x) * h^3/6 + O(h^4)))] / h

= lim(h->0) [(cos(x)(1 - cos(h)) + sin(x)sin(h)) / 1]

= cos(x) lim(h->0) [(1 - cos(h))/h] + sin(x) lim(h->0) [sin(h)/h]

= cos(x) * 0 + sin(x) * 1

= sin(x)

Therefore, the derivative of sec(x) with respect to x using the first principles is sin(x).