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hackensack
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I need to find the derivitive of y=2^x using the definition of derivitive.
brout said:"The derivative is lim (f(x+h)- f(x))/h= axlim {(ah-1)/h}. Notice that that is ax time a limit that is independent of x. That is, as long as the derivative exists, it is ax times a constant. The problem is showing that the lim{(ah-1)/h} EXISTS! And then showing that, if a= 2, that limit is ln(2)."
tell me if I'm wrong, but it doesn't seems so hard to determine this limit..
(a^h-1)/h = (exp (h*ln(a) )-1) / h
= ( 1 + h*ln(a) + o(h*ln(a)) - 1 ) / h h->0
= ln(a) + o(ln(a))
so lim (a^h-1)/h = ln(a) ...
The derivative of a function at a specific point is the slope of the tangent line to the curve at that point.
To find the derivative of a function using the definition, you need to use the limit definition of a derivative, which involves taking the limit as the change in x approaches 0.
The limit definition of a derivative is the mathematical expression that represents the slope of the tangent line to a curve at a specific point.
You apply the limit definition by plugging in the given function into the expression and taking the limit as the change in x approaches 0. This will give you the slope of the tangent line at the specified point.
To find the derivative of y=2^x using the definition, you need to apply the limit definition by plugging in the function, taking the limit as the change in x approaches 0, and simplifying the resulting expression. This will give you the slope of the tangent line at any point on the curve y=2^x.