Tangent lines Find an equation of the line tangent to the graph of f at the given point.
f(x) = sec⁻¹(e^x); (ln 2,π/3)

Verified step by step guidance

1. First, understand that the problem requires finding the equation of the tangent line to the function f(x) = sec^(-1)(e^x) at the point (ln 2, π/3). The general form of a tangent line equation is y - y1 = m(x - x1), where m is the slope of the tangent line and (x1, y1) is the given point.
2. To find the slope m of the tangent line, we need to compute the derivative of f(x) = sec^(-1)(e^x). Recall that the derivative of sec^(-1)(u) with respect to x is 1 / (|u| * sqrt(u^2 - 1)) * du/dx. Here, u = e^x, so we need to apply the chain rule.
3. Calculate the derivative of u = e^x, which is du/dx = e^x. Substitute this into the derivative formula for sec^(-1)(u) to get f'(x) = 1 / (|e^x| * sqrt((e^x)^2 - 1)) * e^x.
4. Evaluate the derivative at the given point x = ln 2. Substitute x = ln 2 into f'(x) to find the slope m of the tangent line. This involves calculating e^(ln 2) and simplifying the expression.
5. Finally, use the point-slope form of the line equation with the point (ln 2, π/3) and the slope m found in the previous step to write the equation of the tangent line. Substitute x1 = ln 2, y1 = π/3, and m into the equation y - y1 = m(x - x1).

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point, which represents the instantaneous rate of change of the function.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. For the function f(x) = sec^−1(e^x), finding the derivative is essential to determine the slope of the tangent line at the specified point.

Inverse Functions

Inverse functions reverse the effect of the original function. For example, if f(x) = sec^−1(x), then its inverse would be sec(x). Understanding how to differentiate inverse functions is crucial, as it involves applying the chain rule and recognizing the relationship between a function and its inverse when calculating derivatives.