Derivative of 9lnx

The derivative of 9ln(x) is denoted as d/dx (9ln(x)) or (9ln(x))', and its value is 9/x. This derivative indicates how the function 9ln(x) changes in response to variations in x, highlighting its differentiability within its domain.

Key concepts include:

Logarithmic Function: ln(x) is the natural logarithm of x.

Constant Multiple Rule: Used for differentiating a constant multiplied by a function.

Reciprocal Function: The derivative of ln(x) is 1/x.

The derivative of 9ln(x) is expressed as d/dx (9ln(x)) or (9ln(x))'. The formula used to differentiate 9ln(x) is: d/dx (9ln(x)) = 9/x

This formula is applicable for all x > 0, as ln(x) is only defined for positive x.

We can derive the derivative of 9ln(x) using proofs. To demonstrate this, we will apply differentiation rules and logarithmic identities. Several methods can be employed, such as:

1. By First Principle
2. Using Constant Multiple Rule
3. Using Chain Rule

We will now show that differentiating 9ln(x) results in 9/x using the methods mentioned:

By First Principle

The derivative of 9ln(x) can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

To find the derivative of 9ln(x) using the first principle, consider f(x) = 9ln(x). Its derivative can be expressed as the following limit: f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 9ln(x), we write f(x + h) = 9ln(x + h).

Substituting these into equation (1), f'(x) = limₕ→₀ [9ln(x + h) - 9ln(x)] / h = 9limₕ→₀ [ln(x + h) - ln(x)] / h = 9limₕ→₀ [ln((x + h)/x)] / h = 9limₕ→₀ ln[(1 + h/x)] / h

Using the limit property, limₕ→₀ ln(1 + u)/u = 1, f'(x) = 9 * 1/x Hence, f'(x) = 9/x, proved.

Using Constant Multiple Rule

To prove the differentiation of 9ln(x) using the constant multiple rule, Consider the constant 9 and the function ln(x). Using the rule: d/dx [c·f(x)] = c·f'(x)

Here, c = 9 and f(x) = ln(x), f'(x) = d/dx [ln(x)] = 1/x Thus, d/dx [9ln(x)] = 9 * 1/x = 9/x.

Using Chain Rule

The chain rule can also be used to differentiate 9ln(x). Consider y = 9ln(x) Let u = ln(x), then y = 9u

By chain rule: dy/dx = dy/du * du/dx dy/du = 9 and du/dx = 1/x

Therefore, dy/dx = 9 * 1/x = 9/x.

When a function is differentiated multiple times, the resulting derivatives are higher-order derivatives. Higher-order derivatives can be complex, similar to understanding a car's speed (first derivative) and its acceleration (second derivative). Higher-order derivatives offer insights into functions like 9ln(x).

For the first derivative, we write f′(x), indicating the slope or rate of change at a point. The second derivative, f′′(x), is derived from the first derivative. The third derivative, f′′′(x), follows from the second derivative, continuing this pattern.

For the nth derivative of 9ln(x), we denote it as fⁿ(x), providing insights into the rate of change for each derivative order.

When x is 0, the derivative is undefined as ln(x) is not defined for non-positive x. When x is 1, the derivative of 9ln(x) = 9/1 = 9.

• Derivative: The derivative of a function measures how it changes in response to a change in x.

• Logarithmic Function: A function involving the logarithm, typically represented as ln(x).

• Constant Multiple Rule: A differentiation rule where a constant is factored out before differentiating a function.

• Chain Rule: A rule for differentiating composite functions, involving the differentiation of both outer and inner functions.

• Quotient Rule: A rule used for differentiating a function that is the quotient of two other functions.