Find the slope of the line tangent to the graph of y = tan^−1 x at x= −2.

Verified step by step guidance

1. First, understand that the slope of the tangent line to the graph of a function at a given point is given by the derivative of the function evaluated at that point.
2. The function given is y = tan^−1(x), which is the inverse tangent function. We need to find its derivative with respect to x.
3. The derivative of y = tan^−1(x) is given by the formula: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math>. This formula comes from the differentiation of the inverse tangent function.
4. Now, substitute x = -2 into the derivative formula to find the slope of the tangent line at this point: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><msup><mo>(</mo><mn>-2</mn><mo>)</mo><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math>.
5. Simplify the expression obtained in the previous step to find the numerical value of the slope. This will give you the slope of the tangent line to the graph at x = -2.

Key Concepts

Derivative

The derivative of a function at a given point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, finding the slope of the tangent line involves calculating the derivative of the function y = tan^−1(x) at the specific point x = -2.

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 represents the instantaneous rate of change of the function at that point. In calculus, the slope of the tangent line can be found using the derivative of the function evaluated at the point of interest.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan^−1(x), are the functions that reverse the action of the standard trigonometric functions. For example, tan^−1(x) gives the angle whose tangent is x. Understanding the properties and derivatives of these functions is essential for solving problems involving their slopes and behaviors, particularly when calculating derivatives at specific points.