Final answer:
To differentiate functions with natural logarithms, you use rules for logarithmic differentiation such as the derivative of ℓn(x) being 1/x, and apply additional rules like the product, quotient, and chain rules where necessary, keeping in mind the properties of logarithms.
Step-by-step explanation:
To find the derivative with respect to x (d/dx) of functions containing natural logarithms (ℓn), you apply the rules of differentiation specifically designed for logarithmic functions. The derivative of ℓn(x) is 1/x, under the condition that x > 0. When differentiating more complex logarithmic functions, you may need to use additional differentiation rules such as the product rule, quotient rule, and chain rule in combination with the properties of logarithms.
For example, to differentiate f(x) = ℓn(x²), you would apply the chain rule. The outer function is the ℓn, and the inner function is x². The derivative is f'(x) = 1/(x²) × 2x, which simplifies to 2/x.
Remember that logarithms follow specific rules, like the fact that the logarithm of a division is the difference between the logarithms of the numerator and the denominator. Also, logarithms are exponents, so operations involving them adhere to the same rules as operations involving exponents.