I need a step-by-step limit definition of derivative for y = ln(x).
To find the limit definition of the derivative for the function y = ln(x), we need to evaluate the following limit:
lim(h->0) [ln(x+h) - ln(x)] / h
Here's a step-by-step breakdown of how to arrive at the derivative using this limit definition:
Step 1: Start with the given function, y = ln(x).
Step 2: Write down the limit expression for the derivative:
lim(h->0) [ln(x+h) - ln(x)] / h
Step 3: Apply the logarithmic property, ln(a) - ln(b) = ln(a/b):
lim(h->0) ln[(x+h)/x] / h
Step 4: Use the logarithmic property, ln(a/b) = ln(a) - ln(b):
lim(h->0) [ln(x+h) - ln(x)] - ln(x) / h
Step 5: Apply the limit property, lim(a - b) = lim(a) - lim(b):
lim(h->0) ln(x+h) - lim(h->0) ln(x) / h
Step 6: Evaluate the limits:
lim(h->0) ln(x+h) = ln(x), since ln(x) is continuous.
lim(h->0) ln(x) = ln(x), since it is a constant.
Step 7: Simplify the expression:
ln(x) - ln(x) / h
Step 8: Simplify further:
0 / h = 0
Step 9: The final limit is 0, and therefore, the derivative of y = ln(x) is 0.
So, the derivative of y = ln(x) is dy/dx = 0.