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Calculus

University

Calculus (University Level)

Derivative of ln(ax)

Compute the derivative of the natural logarithm ln(ax) at a given positive x.

Standard Formula

d/dx[ln(ax)] = 1/x

Interactive Sandbox

Live Reactive

Adjust values in real-time and observe mathematical cause-and-effect

Coefficient (must be > 0)(Real number)

0.120

Value of x (must be > 0)(Real number)

120

Live EquationFormula Proof

d/dx[ln(ax)] = 1/2 = 0.5

Tangent Line dy/dx

Calculated Solution

0.5

Direct computational model evaluating Derivative of ln(ax).

Variable Mechanics

Roles, constraints & impacts

2 Params

Coefficient (must be > 0)

ImpactPrimary formula parameter

ConstraintReal number

Value of x (must be > 0)

ImpactPrimary formula parameter

ConstraintReal number

At a Glance

Gotchas, units & real-world use

Essential

Formula Nature & Mechanics

Direct computational model evaluating Derivative of ln(ax).

Common Mistakes & Traps

Ensure all input measurements share consistent units before calculating.
Check for denominator values approaching zero to prevent undefined division.
Follow standard order of operations (PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract).

Dimensional Analysis & Units

Direct evaluation matching formula variable dimensions.

Real-World & Industry Application

Widely utilized across modern calculus applications, academic analysis, and engineering estimations.

Why the derivative of ln(ax) is 1/x

Chain rule: d/dx[ln(ax)] = (1 / ax) · a = 1/x. The a cancels.

Log rule: ln(ax) = ln(a) + ln(x). ln(a) is a constant with derivative 0, leaving d/dx[ln(x)] = 1/x.

The function ln(ax) is only defined where ax > 0, so for a > 0 you need x > 0.

📖 Beginner's guideDerivative of ln(x) and ln(ax): Easy GuideThe derivative of ln(ax) is 1/x for any positive a, just like ln(x). See why a cancels, examples like ln(3x+1), and use a free calculator with steps.

Worked Example: Derivative of ln(ax)

Given:

a = 1
x = 2

d/dx[ln(x)] = 1/x
At x=2: 1/2 = 0.5

Answer: 0.5

Frequently Asked Questions

What is the Derivative of ln(ax) formula?

d/dx[ln(ax)] = 1/x. Compute the derivative of the natural logarithm ln(ax) at a given positive x.

How do I calculate Derivative of ln(ax)?

Enter Coefficient (must be > 0) (a), Value of x (must be > 0) (x) into the calculator. It applies d/dx[ln(ax)] = 1/x and shows every step of the working.

What is the derivative of ln(ax)?

1/x, for any non-zero constant a. The constant cancels out.

What is the derivative of ln(ax + b)?

a / (ax + b), by the chain rule.

Related Calculus Calculators

Power Rule for Derivativesd/dx(xⁿ) = n × x^(n-1)Definite Integral (Power Rule)∫[a to b] x^n dx = [x^(n+1)/(n+1)]|[a to b]Partial Derivative∂f/∂x = lim[h→0] (f(x+h,y) - f(x,y))/h Chain Rule (Derivative)d/dx[f(g(x))] = f'(g(x)) · g'(x)Product Rule (Derivative)d/dx[f·g] = f'·g + f·g'Quotient Rule (Derivative)d/dx[f/g] = (f'·g − f·g') / g²

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