Question

if sinusoidal functions is not even then its derivative is even. true or false?

Answer 1

True. The derivative of any odd function is always even.

Answer 2

And for the proof of such we can use the chain rule and definition of an odd function:

f is odd if the following is true: f(x) = - f(-x)

let z = -x

dz/dx = -1 [equivalently z' = -1]

Make a z substitution in the definition of the odd function f(x)

f(x) = - f(z)

differentiate using chain rule:

df(x)/dx =-[ df(z)/dz ] * [dz/dx] {equivalently, f '(x) = - f '(z) z' }

df(x)/dx = -[df(z)/dz] * (-1)

so, df(x)/dx = df(z)/dz

f '(x) = f '(z) substitute for z = -x

f '(x) = f '(-x) satisfies the definition of an even function and since we learned in lecture that f ' is indeed a function it is also an even function. So, derivatives of odd functions are even.

Answer 3

given f(x)=-f(-x) =>odd
diff both side
f'(x)=-(-f'(-x))
or f'(x)=f'(-x) => even
so odd sinusoid has even derv

Answer 4

and vice-versa