y = (sin x) / (1 + tan x)

Is there a question here?

Sorry

y'=

just use the quotient rule. If

y = u/v
y' = (vu'-uv')/v^2

y' = [(1+tanx)(cosx) - (sinx)(sec^2 x)]/(1+tanx)^2
= ((cosx - (sinx)(tan^2 x))/(1+tanx)^2

The equation you provided is y = (sin x) / (1 + tan x). To understand this equation better, let's break it down:

1. Start with the denominators:
a. The denominator of the fraction is (1 + tan x). It consists of two parts: 1 and tan x.
b. The number 1 represents a constant value of 1.
c. The tan x is the tangent function of x. Tangent is the ratio of the sine function to the cosine function (tan x = sin x / cos x).

2. Now, let's consider the numerator:
a. The numerator is sin x, which represents the sine function of x. The sine function calculates the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.

Putting it all together, the equation calculates the ratio of sin x to the sum of 1 and tan x.

To solve this equation, there are a few approaches you can take:

1. Analytical Approach:
a. Simplify the equation by substituting tan x with sin x / cos x.
b. Multiply both sides of the equation by (1 + tan x) to eliminate the denominator.
c. Simplify the equation further by using trigonometric identities and algebraic manipulations.
d. Solve for y to obtain the final expression in terms of x.

2. Numerical Approach:
a. You can approximate the values of y for different values of x by substituting x into the equation.
b. Use a calculator or software program that supports trigonometric functions to calculate the value of y for specific x values.

Remember to always check for any restrictions on x that might result in undefined values for the equation (e.g., dividing by zero or taking the square root of a negative number if applicable).