Derivative of ln(1+2x)

We now understand the derivative of ln(1+2x). It is commonly represented as d/dx (ln(1+2x)) or (ln(1+2x))', and its value is 2/(1+2x). The function ln(1+2x) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Logarithmic Function: (ln(x) is the natural logarithm function). Chain Rule: Rule for differentiating composite functions like ln(1+2x). Reciprocal Rule: Used when differentiating functions involving reciprocals.

The derivative of ln(1+2x) can be denoted as d/dx (ln(1+2x)) or (ln(1+2x))'. The formula we use to differentiate ln(1+2x) is: d/dx (ln(1+2x)) = 2/(1+2x)

We can derive the derivative of ln(1+2x) using proofs. To show this, we will use the rules of differentiation along with the chain rule. There are several methods we use to prove this, such as: By Chain Rule By Logarithmic Differentiation We will now demonstrate that the differentiation of ln(1+2x) results in 2/(1+2x) using the above-mentioned methods: Using Chain Rule To prove the differentiation of ln(1+2x) using the chain rule, Consider u = 1+2x. Then, ln(1+2x) = ln(u). The derivative of ln(u) with respect to u is 1/u. By using the chain rule, the derivative of ln(1+2x) with respect to x is: d/dx [ln(u)] = (1/u) · du/dx = (1/(1+2x)) · 2 = 2/(1+2x) Hence, proved. Using Logarithmic Differentiation Consider y = ln(1+2x). By implicitly differentiating, we have: d/dx [ln(y)] = d/dx [ln(1+2x)] Using the chain rule, dy/dx = (1/(1+2x)) · 2 dy/dx = 2/(1+2x) Thus, proved.

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky. To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes. Higher-order derivatives make it easier to understand functions like ln(1+2x). For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point. The second derivative is derived from the first derivative, which is denoted using f′′(x). Similarly, the third derivative, f′′′(x), is the result of the second derivative and this pattern continues. For the nth Derivative of ln(1+2x), we generally use fⁿ(x) for the nth derivative of a function f(x) which tells us the change in the rate of change. (continuing for higher-order derivatives).

When x is such that 1+2x = 0, the derivative is undefined because ln(1+2x) is undefined there. When x is 0, the derivative of ln(1+2x) = 2/(1+2*0) = 2.

Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Logarithmic Function: The natural logarithm function is represented as ln(x). Chain Rule: A rule used to differentiate composite functions. Quotient Rule: A rule used to differentiate functions that are ratios of other functions. Domain: The set of all possible input values (x-values) for which a function is defined.