a question about derivative.

[XIE-Yao]

http://dec41.user.srcf.net/h/IA_M/differential_equations/1_1

f'(2x) = d f / d(2x) ?

is this right?

[jftsang]

In general, no. f'(2x) means "evaluate the function f' at the value 2x. The function f' is "the derivative of f wrt its argument", or "the derivative of f(t) wrt t, where t is a dummy variable".

In the Leibniz notation df/dx or df/d(2x) you do not specify where the expression is being evaluated.

f(t) = sin(t)
f'(t) = cos(t)
f'(2x) = cos(2x)  # evaluate at t = 2x

but

df/d(2x) = df/dx * dx/d(2x) = 1/2 df/dx

introduces a factor of 1/2 out front.

If however

g(y) = sin(2y)

then

g'(y) = 2 cos(2y)

[XIE-Yao]

exactly what i'm thinking.
cannot agree with you any more.

In general, no. f'(2x) means "evaluate the function f' at the value 2x. The function f' is "the derivative of f wrt its argument", or "the derivative of f(t) wrt t, where t is a dummy variable".

In the Leibniz notation df/dx or df/d(2x) you do not specify where the expression is being evaluated.

f(t) = sin(t)
f'(t) = cos(t)
f'(2x) = cos(2x)  # evaluate at t = 2x

but

df/d(2x) = df/dx * dx/d(2x) = 1/2 df/dx

introduces a factor of 1/2 out front.

If however

g(y) = sin(2y)

then

g'(y) = 2 cos(2y)