Question

differentiate the following functions with respect to X
y=
$\frac{ \sqrt{x + 1} }{ \sqrt{x - 1} }$
please answer correct ​

Answer 1

Answer:

We have already studied how to find equations of tangent lines to functions and the rate of change of a function at a specific point. In all these cases we had the explicit equation for the function and differentiated these functions explicitly. Suppose instead that we want to determine the equation of a tangent line to an arbitrary curve or the rate of change of an arbitrary curve at a point. In this section, we solve these problems by finding the derivatives of functions that define y implicitly in terms of x.

Implicit Differentiation

In most discussions of math, if the dependent variable y is a function of the independent variable x, we express y in terms of x. If this is the case, we say that y is an explicit function of x. For example, when we write the equation y=x^2+1, we are defining y explicitly in terms of x. On the other hand, if the relationship between the function y and the variable x is expressed by an equation where y is not expressed entirely in terms of x, we say that the equation defines y implicitly in terms of x. For example, the equation y-x^2=1 defines the function y=x^2+1 implicitly.

Answer 2

Answer:

How do you differentiate the following function with respect to x: y = (x -2) (x +3)?

The easier way:

Multiply the terms, giving y = x² + x - 6.

Differentiate term by term: y’ = 2x + 1 - 0 = 2x + 1.

The harder way:

Use the product rule to differentiate the product of functions (x-2)(x+3).

Let u = x-2 and v = x+3.

d/dx [u·v] = u·v’ + v·u’, so dy/dx = (x-2)(x+3)’ + (x+3)(x-2)’

= (x-2)(1) + (x+3)(1)

= x - 2 + x + 3

= 2x + 1.