I have equation for tangent line: $y=x+1$ and the tangent line is through point $(x_0,y_0)$ whose coordinate is $(0,1)$ therefore from this formula $m=1=e^x0=e^0$ (the slope of the tangent line) and thus the derivative of the function we are searching for is: $e^x$ to have the function we are searching for,we need to compute $\int e^x~dx= e^x+C$ and we have graph function is through point $(0,1)$ so the constant is just zero therefore the function we are searching for is $e^x$
But is this correct way? How do I know it is the slope of function e^x or some other function?
Thank you so much!
Best Answer
I'll just give a few things, which you should prove:
Every function staisfying the following two equations would give you a correct solution: $$f(0) = 1 \\ f'(0)=1$$
For example calculate the tangent line at the point $(0,1)$ of the function $f(x) = (x+\frac{1}{2})^2 + \frac{3}{4}$.