Final answer:
To find the equation of a tangent line to the curve y = ln(x) at a given point, differentiate the function to find the slope and use the point-slope form to form the equation, simplifying it to get the final form.
Step-by-step explanation:
To find an equation of the tangent line to the curve y = ln(x) at a given point, we need to follow several steps involving differentiation and application of the point-slope form of a line.
Steps to Find the Tangent Line
- First, differentiate the equation y = ln(x) to find the slope of the tangent line. The derivative of ln(x) with respect to x is 1/x.
- Next, calculate the slope m of the tangent line using the x-coordinate of the given point by substituting it into the derivative.
- With the slope m known and the given point (x1, y1), use the point-slope form of a line: y - y1 = m(x - x1).
- Finally, simplify the equation to obtain the desired tangent line equation in the form y = mx + b, where b is the y-intercept.
As an example, to find the tangent line at the point (1, 0), which lies on the curve y = ln(x), we would calculate the slope at x = 1 giving us m = 1/1 = 1. Thus, the equation of the tangent line would be y - 0 = 1(x - 1), which simplifies to y = x - 1.