How do I us the Limit definition of derivative on #f(x)=tan(x)#?

Answer 1

By Limit Definition,

#f'(x)=lim_{h to 0}{tan(x+h)-tanx}/h#
by the trig identity: #tan(alpha+beta)={tan alpha +tan beta}/{1-tan alpha tan beta}#,
#=lim_{h to 0}{{tan x+tan h}/{1-tan x tan h}-tan x}/h#

by taking the common denominator,

#=lim_{h to 0}{{tan x + tan h-(tan x - tan^2x tan h)}/{1-tan x tan h}}/h#
by cancelling out #tan x#'s,
#=lim_{h to 0}{{tan h +tan^2x tan h)/{1-tan x tan h}}/h#
by factoring out #tan h#,
#=lim_{h to 0}({tan h}/h cdot {1+tan^2 x}/{1-tan x tan h})#
by #tan h ={sin h}/{cos h}# and #1+tan^2x=sec^2x#,
#=lim_{h to 0}({sin h}/h cdot 1/{cos h} cdot {sec^2x}/{1-tan x tan h})#
by #lim_{h to 0}{sin h}/h=1#,
#=1 cdot 1/1 cdot {sec^2x}/1=sec^2 x#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To use the limit definition of the derivative on ( f(x) = \tan(x) ), follow these steps:

  1. Recall the definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

  2. Substitute ( f(x) = \tan(x) ) into the definition: [ f'(x) = \lim_{h \to 0} \frac{\tan(x + h) - \tan(x)}{h} ]

  3. Use the tangent addition formula: [ \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} ]

  4. Apply the tangent addition formula to the numerator: [ \tan(x + h) = \frac{\tan(x) + \tan(h)}{1 - \tan(x)\tan(h)} ]

  5. Substitute this expression into the derivative definition: [ f'(x) = \lim_{h \to 0} \frac{\frac{\tan(x) + \tan(h)}{1 - \tan(x)\tan(h)} - \tan(x)}{h} ]

  6. Simplify the expression: [ f'(x) = \lim_{h \to 0} \frac{\tan(x) + \tan(h) - \tan(x)(1 - \tan(x)\tan(h))}{h(1 - \tan(x)\tan(h))} ]

  7. Combine like terms and factor out ( \tan(x) ): [ f'(x) = \lim_{h \to 0} \frac{\tan(h)}{h(1 - \tan(x)\tan(h))} ]

  8. As ( h ) approaches 0, ( \tan(h) ) approaches ( 0 ): [ f'(x) = \frac{0}{1 - \tan(x)(0)} = 0 ]

Therefore, the derivative of ( f(x) = \tan(x) ) is ( f'(x) = 0 ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7