what is the derivative of
ln(y) = x
To find the derivative of ln(y) = x, we can follow these steps:
Step 1: Rewrite the equation in exponential form.
Taking the exponential of both sides, we have:
e^(ln(y)) = e^x
Step 2: Simplify the left side.
Since e^(ln(y)) is the inverse function of ln(y), it cancels out, leaving us with just y.
So, now we have:
y = e^x
Step 3: Take the derivative of both sides.
To find the derivative of y = e^x, we can use the chain rule, which states that for a composite function y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x).
In this case, we have y = e^x, where f(u) = e^u and g(x) = x.
The derivative of e^x with respect to x is simply e^x.
So, dy/dx = f'(g(x)) * g'(x) = e^x * 1 = e^x.
Therefore, the derivative of ln(y) = x is dy/dx = e^x.