When you want to find the derivatives of natural log functions, there are some easy ways to make it simpler.
First, remember that the derivative of the natural logarithm function, which is written as ( f(x) = \ln(x) ), is:
[ f'(x) = \frac{1}{x}. ]
But if you have more complicated log functions, you need some extra tricks to help you.
If your function includes a natural log that’s part of a bigger function, like ( f(x) = \ln(g(x)) ), you should use the Chain Rule. This rule helps us apply the derivative properly:
[ f'(x) = \frac{g'(x)}{g(x)}. ]
Example: If ( f(x) = \ln(3x^2 + 2) ), using the Chain Rule gives you:
[ f'(x) = \frac{6x}{3x^2 + 2}. ]
If your log function is multiplied by another function or divided by one, use the Product or Quotient Rule first. After that, you can differentiate the log functions.
Using some rules about logarithms can also help make finding derivatives easier. For example, the rule ( \ln(a/b) = \ln(a) - \ln(b) ) can break complex expressions into simpler parts.
Example: For ( f(x) = \ln\left(\frac{x^2 + 1}{x}\right) ):
Start by using the property:
[ f'(x) = \frac{2x}{x^2 + 1} - \frac{1}{x} = \frac{2x^2 - (x^2 + 1)}{x(x^2 + 1)} = \frac{x^2 - 1}{x(x^2 + 1)}. ]
Using these techniques will help you handle the derivatives of natural log functions much more easily!
When you want to find the derivatives of natural log functions, there are some easy ways to make it simpler.
First, remember that the derivative of the natural logarithm function, which is written as ( f(x) = \ln(x) ), is:
[ f'(x) = \frac{1}{x}. ]
But if you have more complicated log functions, you need some extra tricks to help you.
If your function includes a natural log that’s part of a bigger function, like ( f(x) = \ln(g(x)) ), you should use the Chain Rule. This rule helps us apply the derivative properly:
[ f'(x) = \frac{g'(x)}{g(x)}. ]
Example: If ( f(x) = \ln(3x^2 + 2) ), using the Chain Rule gives you:
[ f'(x) = \frac{6x}{3x^2 + 2}. ]
If your log function is multiplied by another function or divided by one, use the Product or Quotient Rule first. After that, you can differentiate the log functions.
Using some rules about logarithms can also help make finding derivatives easier. For example, the rule ( \ln(a/b) = \ln(a) - \ln(b) ) can break complex expressions into simpler parts.
Example: For ( f(x) = \ln\left(\frac{x^2 + 1}{x}\right) ):
Start by using the property:
[ f'(x) = \frac{2x}{x^2 + 1} - \frac{1}{x} = \frac{2x^2 - (x^2 + 1)}{x(x^2 + 1)} = \frac{x^2 - 1}{x(x^2 + 1)}. ]
Using these techniques will help you handle the derivatives of natural log functions much more easily!