Derivative of ln
The derivative of the natural logarithm function, ln(x), can be found using the rules of differentiation
The derivative of the natural logarithm function, ln(x), can be found using the rules of differentiation. Let’s derive it step by step:
Step 1: Start with the definition of the natural logarithm:
ln(x) = y —> e^y = x
Step 2: Take the derivative of both sides of the equation with respect to x:
d/dx(e^y) = d/dx(x)
Step 3: To find the derivative of e^y, we can use the chain rule. The chain rule states that if we have a function g(h(x)), then its derivative is given by g'(h(x)) * h'(x). In this case, g(u) = e^u and h(x) = y.
Therefore, d/dx(e^y) = d/du(e^u) * d/dx(y)
Step 4: Since y is just a function of x, we can rewrite d/dx(y) as dy/dx. So, our equation becomes:
e^y * dy/dx = 1
Step 5: Now, solve for dy/dx:
dy/dx = 1 / e^y
Step 6: Substitute the value of y from Step 1 into the equation:
dy/dx = 1 / e^(ln(x))
Step 7: Since e^(ln(x)) is equal to x, we simplify further:
dy/dx = 1 / x
Therefore, the derivative of ln(x) is 1/x.
In summary, the derivative of ln(x) with respect to x is 1/x.
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