Given $f(x) = 5^{3x}$. Find $f'(x)$ using definition of a derivative.
The definition of the derivative of $f(x)$ is
$f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$
The derivative of $f(x) = 5^{3x}$ is
$f'(x) = \lim_{h \to 0} \dfrac{5^{3(x + h)} - 5^{3x}}{h}=\lim_{x \to 0} \dfrac{5^{3x}5^{3h} - 5^{3x}}{h}=\lim_{x \to 0} \dfrac{5^{3x}\left(5^{3h} - 1\right)}{h}$
And this is where I get stuck. Is there any way to proceed from here appropriately?
What you need to show is that:
$$
\lim_{h\to 0} \frac{5^{3h}-1}{h}=\lim_{h\to 0} \frac{125^{h}-1}{h}=\ln(125)
$$
Note, we have the following:
$$\lim_{x\to 0} \frac{\log_a(1+x)}{x} = \lim_{x\to 0} \frac{\log_e(1+x)}{x\log_e{a}}=\lim_{x\to0} \frac{\sum_{i=1}^{\infty} (-1)^{n+1} \frac{x^{n}}{n}}{x\log{a}} =\lim_{x\to 0} \frac{1+\sum_{i=2}^{\infty} (-1)^{n+1} \frac{x^{n}}{n}}{\log{a}} = \frac{1}{\log{a}} = \frac{1}{\frac{\log_a{a}}{\log_{a}{e}}} = \log_{a}{e}$$
Now set $y=a^x -1.$ Hence $a^x = 1+y$ and
$$
x = \log_a(1+y)
$$
As $x\to 0; y\to 0$
Therefore:
$$
\lim_{x\to 0}\frac{a^x-1}{x}=\lim_{y\to 0}\frac{y}{\log_a(1+y)}=\log_e a
$$
This is taken from my answer here: Proving that $\lim_{h\to 0 } \frac{b^{h}-1}{h} = \ln{b}$
You might look at some other approaches there.
