It follows fromZedCar said:Homework Statement
Differentiate with respect to x
y = tan^(-1) x
Homework Equations
The Attempt at a Solution
ANSWER in book:
y = tan^(-1) x
x = tan y
dx/dy = sec^(2) y = (1 + x^2)
dy/dx = 1 / (1 + x^2)
How is it known that sec^(2) y = (1 + x^2)
Does it follow from an identity?
Differentiating with respect to x means finding the derivative of a function with respect to the independent variable x. This involves calculating the rate of change of the function with respect to small changes in x.
Differentiating with respect to x is important because it allows us to analyze the behavior of a function and determine its rate of change at different points. This is useful in many fields of science, including physics, engineering, and economics.
To differentiate with respect to x, we use the rules of differentiation, such as the power rule, product rule, and chain rule. These rules allow us to find the derivative of a function by breaking it down into simpler components and applying specific formulas.
Sure, let's take the function f(x) = 3x^2 + 2x. To differentiate this function with respect to x, we first use the power rule to get f'(x) = 6x + 2. This means that the rate of change of f(x) with respect to x is 6x + 2 at any given point.
When we differentiate a function with respect to x, we get its derivative, which represents the slope of the function at any given point. By finding the points where the derivative is equal to 0, we can determine the extrema (maximum and minimum points) of the function. This is useful in optimization problems and in determining the critical points of a function.