Derivative of Logarithmic Function Calculator

Enter the constants a and k together with an x value in the real domain. The calculator applies the chain rule, simplifies the derivative, checks that the logarithm is defined, and returns a clear symbolic expression plus its numerical value at the chosen point.

Describe the logarithmic function

Start by identifying the three numbers in the problem. The coefficient a multiplies the entire logarithm, while k multiplies x inside the logarithm. The last input, x, is the point where the derivative will be evaluated. For example, a function written as six times ln of four x has a equal to 6 and k equal to 4. If the requested point is 3, enter x as 3. All three inputs must be finite numbers, and k cannot be zero. The calculator works with positive or negative values when the real logarithm permits them. In particular, k and the evaluation point may both be negative because their product is then positive. Keeping the outer coefficient separate from the inner scale factor helps prevent a common setup error: a affects the magnitude of the derivative, while k determines the real domain of the original function. Once the values are entered, the tool displays the interpreted function so you can confirm that the mathematical structure matches your exercise before using the evaluated result.

Apply the chain rule and simplify

For f(x) = a ln(kx), begin with the standard rule that the derivative of ln(u) is u prime divided by u. Here the inner expression u is kx, so its derivative is k. Multiplying by the outside coefficient gives a times k divided by kx. Because k is nonzero, it cancels, leaving the simplified derivative a divided by x. This cancellation explains why the numerical derivative does not depend on the size of k, even though k still matters to the original function and its domain. The calculator performs this simplification directly and reports the derivative in a readable symbolic form. This is a useful check against mistakenly writing a divided by kx or forgetting the inner derivative. The rule applies only where the original real-valued logarithm exists. It is therefore not enough for x merely to be nonzero: the product kx must be positive. The tool validates that condition before evaluating the simplified expression, preserving the domain restrictions of the function that was actually supplied.

Evaluate and interpret the result

After simplifying the derivative to a divided by x, substitute the requested point and divide. The returned derivative value is the instantaneous rate of change of the original logarithmic function at that point, or equivalently the slope of its tangent line there. A positive result means the function is increasing locally, while a negative result means it is decreasing locally. A value of zero occurs when a is zero, in which case the original function is constant wherever the logarithm is defined. The output includes the interpreted function, the simplified derivative, the evaluation point, and the numerical derivative value, making it suitable for checking handwritten work or feeding a deterministic result into another calculation. Remember that the derivative formula alone does not erase the original domain. Although a divided by x is algebraically defined for every nonzero x, an evaluation is valid for this problem only when ln(kx) is real. Invalid zero, nonfinite, or out-of-domain inputs are rejected instead of producing a misleading number. Automated API requests use the published base price of $0.002 per calculation.