Answer:
METHOD I:
(by using the first principle of differentiation)
We have the "Limit definition of Derivatives":
[tex]\boxed{\mathsf{f'(x)= \lim_{h \to 0} \{\frac{f(x+h)-f(x)}{h} \} ....(i)}}[/tex]
Here, f(x) = sec x, f(x+h) = sec (x+h)
- Substituting these in eqn. (i)
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \{\frac{sec(x+h)-sec(x)}{h} \} }[/tex]
- sec x can be written as 1/ cos(x)
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{1}{h} \{\frac{1}{cos(x+h)} -\frac{1}{cos(x)} \} }[/tex]
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{1}{h} \{\frac{cos(x)-cos(x+h)}{cos(x)cos(x+h)} \} }[/tex]
- By Cosines sum to product formula, i.e.,
[tex]\boxed{\mathsf{cos\:A-cos\:B=-2sin(\frac{A+B}{2} )sin(\frac{A-B}{2} )}}[/tex]
=> cos(x) - cos(x+h) = -2sin{(x+x+h)/2}sin{(x-x-h)/2}
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{2sin(x+\frac{h}{2} )}{cos(x+h)cos(x)}\:.\: \lim_{h \to 0} \frac{sin(\frac{h}{2} )}{h} }[/tex]
- I shifted a 2 from the first limit to the second limit, since the limits ar ein multiplication this transmission doesn't affect the result
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{sin(x+\frac{h}{2} )}{cos(x+h)cos(x)}\:.\: \lim_{h \to 0} \frac{2sin(\frac{h}{2} )}{h} }[/tex]
- 2/ h can also be written as 1/(h/ 2)
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{sin(x+\frac{h}{2} )}{cos(x+h)cos(x)}\:.\: \lim_{h \to 0} \frac{1\times sin(\frac{h}{2} )}{\frac{h}{2} } }[/tex]
- We have limₓ→₀ (sin x) / x = 1.
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{sin(x+\frac{h}{2} )}{cos(x+h)cos(x)}\:.\: 1 }[/tex]
Substituting 0 for h and h/ 2
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{sin(x+0)}{cos(x+0)cos(x)} }[/tex]
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{sin(x)}{cos(x)cos(x)} }[/tex]
[tex]\implies \mathsf{f'(x)= \lim_{h \to 0} \frac{sin(x)}{cos(x)}\times \frac{1}{cos x} }[/tex]
- sin x/ cos x is tan x whereas 1/ cos (x) is sec (x)
[tex]\implies \mathsf{f'(x)= tan(x)\times sec(x) }[/tex]
Hence, we got
[tex]\underline{\mathsf{\overline{\frac{d}{dx} (sec(x))=sec(x)tan(x)}}}[/tex]
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METHOD II:
(by using other standard derivatives)
[tex] \boxed{ \mathsf{ \frac{d}{dx} ( \sec \: x) = \sec x \tan x }}[/tex]
- sec x can also be written as (cos x)⁻¹
We have a standard derivative for variables in x raised to an exponent:
[tex] \boxed{ \mathsf{ \frac{d}{dx}(x)^{n} = n(x)^{n - 1} }}[/tex]
Therefore,
[tex] \mathsf{ \frac{d}{dx}( \cos x)^{ - 1} = - 1( \cos \: x) ^{( - 1 - 1} } \\ \implies \mathsf{\ - 1( \cos \: x) ^{- 2 }}[/tex]
- Any base with negative exponent is equal to its reciprocal with same positive exponent
[tex] \implies \: \mathsf{ - \frac{1}{ (\cos x) {}^{2} } }[/tex]
The process of differentiating doesn't just end here. It follows chain mechanism, I.e.,
while calculating the derivative of a function that itself contains a function, the derivatives of all the inner functions are multiplied to that of the exterior to get to the final result.
- The inner function that remains is cos x whose derivative is -sin x.
[tex] \implies \mathsf{ - \frac{1}{ (\cos x )^{2} } \times ( - \sin x) }[/tex]
- cos²x can also be written as (cos x).(cos x)
[tex] \implies \mathsf{ \frac{ \sin x }{ \cos x } \times ( \frac{1}{cos x} ) }[/tex]
- sin x/ cos x is tan x, while 1/ cos x is sec x
[tex] \implies \mathsf{ \tan x \times \sec x }[/tex]
= sec x. tan x
Hence, Proved!
