I am learning Calculus and I ran into the proof regarding the exponential derivative, I haven't quite understood it though, could you guys please help me out with that? The proof goes like that:
$$y = a^x$$ $$ln(y) = ln(a^x)$$ $$ln(y) = x(ln(a))$$
Here comes the part I haven't gotten yet, why is y'/y = lna?
$$ \frac{y'}{y} = lna$$ $$y´= lna(y)$$ $$y´= ln(a)(a^x)$$
Thanks in advance!
On
When you go from $ln(y)=x(ln(a))$, we differentiate both sides. For the left side:
$$\frac{d}{dx}ln(y)$$ Which by the chain rule yields:
$$\frac{d}{dx}y\times\frac{d}{dy}ln(y)$$ $$=y'\times\frac{1}{y}$$ $$=\frac{y'}{y}$$
For the right-hand side, the derivative of $ln(y)=x(ln(a))$ is $ln(a)$ (which can easily be confirmed) and you can then solve what $y'$ is.
$$ \ln (y) = \ln (a^x)\\ \ln (y) = x \ln (a)\\ y´/y = \ln a $$
Where from 2nd to 3rd, you differentiate with respect to x on both sides.
Derivative of $\ln x$ is $1/x$. So using the chain rule, derivative of $\ln y$ is $1/y * dy/dx$. Rearrange and use prime notation for derivative and you get $y´/y$