Hey. I'm having trouble understanding part of the definition of this derivative. Any help will be appreciated.(adsbygoogle = window.adsbygoogle || []).push({});

[tex] f(x)=a^x [/tex]

Using the definition of a derivative, the derivative of the above function is:

[tex]f'(x) = \lim_{h \rightarrow 0}\frac{a^{x+h} + a^x}{h} = [/tex]

[tex] \lim_{h \rightarrow 0} \frac{a^x(a^h - 1)}{h} [/tex]

Since a^x does not depend on h it can be taken outside the limit:

[tex] f'(x) = a^x \lim_{h \rightarrow 0} \frac{a^h-1}{h} [/tex]

Now here is where I get confused. The text tells me that:

[tex] \lim_{h \rightarrow 0} \frac{a^x(a^h - 1)}{h} = f'(0) [/tex] (1)

If that is true then [tex] f'(x) = f'(0)a^x [/tex], but I have no idea why equation 1 is the way it is? How is that limit equal to f'(0)?

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# Definition of d(a^x)/dx

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