Question

Tangent lines Find an equation of the line tangent to the graph of \(f\) at the given point. $$f(x)=\tan ^{-1} 2 x ;(1 / 2, \pi / 4)$$

Short answer

Question: Find the equation of the tangent line to the graph of \(f(x) = \tan^{-1}(2x)\) at \(x = \frac{1}{2}\). Answer: The equation of the tangent line is \(y = x + \frac{\pi}{4} - \frac{1}{2}\).

Step-by-step solution

Find the derivative of f(x)

To find the tangent line's slope, we need to find the derivative of the given function, \(f(x)=\tan^{-1}(2x)\). We can use the chain rule to do this. Recall the derivative of \(\tan^{-1}(u)\) is \(\frac{1}{1+u^2}\), multiplied by the derivative of u. Therefore, the derivative of our function is: $$ f'(x) = \frac{1}{1+(2x)^2} \cdot 2 $$

Evaluate f'(x) at x = 1/2

Now that we have found that \(f'(x) = \frac{2}{1+4x^2}\), we will evaluate the derivative at the given point, \(x=\frac{1}{2}\): $$ f'\left(\frac{1}{2}\right) = \frac{2}{1+4\left(\frac{1}{2}\right)^2} = \frac{2}{1+1} = 1 $$ So, the slope of the tangent line is 1.

Use point-slope form to find the equation of the tangent line

Now that we have the slope and a point on the tangent line, we can use the point-slope form of a linear equation to find the equation of the tangent line, which is given by \(y-y_1=m(x-x_1)\), where \((x_1,y_1)\) is a point on the line, and m is the slope. We know that the point given is \((1/2, \pi/4)\) and slope is 1, so we can plug these values into our formula: $$ y-\frac{\pi}{4} = 1\left(x-\frac{1}{2}\right) $$ And simplifying to get our final equation: $$ y = x+\frac{\pi}{4} - \frac{1}{2} $$

Key concepts

Derivative

Understanding the concept of the derivative is fundamental to calculus and many of its applications. The derivative of a function at a point measures the rate at which the function's value is changing with respect to a change in its input value. In simple terms, it gives us the slope of the tangent line to the function's graph at that point. This is why when calculating the equation of a tangent line, the first step is usually to find the derivative of the function.

The process of finding a derivative is called differentiation, and for different functions, there are various rules and methods to find derivatives. The derivative of basic functions, like polynomials, can be found using simple formulas, while more complex functions may require techniques such as the product rule, quotient rule, or the chain rule. The derivative not only helps in graphing the slope of tangent lines but also serves in optimization problems, motion analysis, and in the understanding of the behavior of functions.

Point-Slope Form

The point-slope form is an equation of a straight line which is particularly useful when we know one point on the line and its slope. The general formula is written as:
\( y - y_1 = m(x - x_1) \)
where \(m\) is the slope of the line, and \(x_1, y_1\) are the coordinates of the known point on the line. This form is highly versatile because, with just two pieces of information, you can immediately write down the equation of the line.

To use this form effectively, substitute the known values into the equation, and then simplify to find the form you need, which is typically the slope-intercept form \(y = mx + b\). In the case of tangent lines in calculus, we first find the slope by taking the derivative and evaluating it at the point of tangency. The result, along with the coordinates of the point, is then substituted into the point-slope form to yield the tangent line's equation.

Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the standard trigonometric functions. They are used to determine angles given the value of the trigonometric function. For example, the inverse of the sine function is the arcsine, denoted as \(\sin^{-1}\) or \(\arcsin\), which yields the angle whose sine is a given number.

The same applies to the tangent function, where the inverse \(\tan^{-1}\) or \(\arctan\) gives the angle whose tangent is the given number. These functions are essential, especially in integration and differentiation involving trigonometric functions because they can help express complex algebraic relationships in geometric terms.

In calculus problems like finding the slope of a tangent line to a function involving an inverse trigonometric function, we often differentiate the function to get an expression that includes these inverse operations.

Chain Rule

The chain rule is a powerful differentiation technique used in calculus to find the derivative of a composite function. In essence, if a function y is composed of another function u, which is in turn a function of x, then the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x.

The chain rule formula looks like this:
\( \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} \)
When dealing with trigonometric functions, particularly inverse trigonometric functions, the chain rule becomes critical. This is because these functions often include expressions within them that are more complex than a simple x.

By applying the chain rule, you can break down the process of differentiation into manageable steps, which makes finding the derivative of even the most intricate functions feasible. Each part of the composition is differentiated individually, then the results are multiplied together to get the final derivative. This method is used in the step-by-step solution above to find the derivative of \(\tan^{-1}(2x)\).