Calculus/Chain Rule

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Expert: Alon Mandes - 10/20/2009


Question
Let f be a differentiable function. use the chain rule to show that :
(a) if f is even then f0 is odd.
(b) if f is odd, then f0 is even.

I really don't understand this. Thank you

Answer
Let's 1st of all review the definitions of even & odd :
Function f(x) is even if :
f(x)=f(-x)
Function f(x) is even if :
f(-x)=-f(x)
Now,

If f(x) is even , that means : f(x)=f(-x) . Now let's derive both sides of the equation:
d/dx { f(x) } = d/dx { f(-x) }
f'(x)*[dx/dx] = f'(-x)*[d(-x)/dx]
f'(x)*1=f'(-x)*(-1)
If we set g=f' then we get :
g(x)=-g(-x)
That means g is odd.

If f(x) is odd , that means : f(-x)=-f(x) . Now let's derive both sides of the equation:
d/dx { f(-x) } = d/dx { -f(x) }
f'(-x)*[d(-x)/dx] = -f'(x)*[dx/dx]
f'(-x)*(-1)=-f'(x)*1
-f'(-x)=-f'(x)
f'(-x)=f'(x)
If we set g=f' then we get :
g(x)=g(-x)
That means g is even.

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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