Derivative of ln x
The derivative of ln x is an important result in calculus
The derivative of ln x is an important result in calculus. To find the derivative of ln x, we use the chain rule. Let’s go step by step:
Step 1: Start with the function f(x) = ln x.
Step 2: Recall that the natural logarithm function ln x is defined as the inverse of the exponential function e^x. In other words, ln x is the number y such that e^y = x.
Step 3: Take the derivative of both sides of the equation e^y = x with respect to x using the chain rule. The chain rule states that if f(g(x)) is a composite function, then its derivative is given by f'(g(x)) * g'(x).
The derivative of e^y with respect to x is e^y * dy/dx because e^y depends on both y and x. In other words, it is the derivative of the outside function multiplied by the derivative of the inside function.
The derivative of x with respect to x is simply 1.
So, we have e^y * dy/dx = 1.
Step 4: Solve for dy/dx, which is the derivative of ln x:
dy/dx = 1 / e^y.
Step 5: Substitute y back in terms of x using the definition of ln x. Since ln x is defined as the number y such that e^y = x, we have y = ln x.
Therefore, dy/dx = 1 / e^(ln x).
Step 6: Simplify the right-hand side of the equation:
Recall that e^(ln x) is equal to x, so we have:
dy/dx = 1 / x.
Therefore, the derivative of ln x with respect to x is 1 / x.
In conclusion, the derivative of ln x is 1 / x.
More Answers:
[next_post_link]