Question

Please give solution with steps differentiate y= $\sqrt{log(x³+1}$ {ignore the A it is x^3}

Answer 1

Answer:

Step-by-step explanation:

Given a function f( x, y) of two variables, its total differential df is defined by the equation

Example 1: If f( x, y) = x 2 y + 6 x – y 3, then

The equation f( x, y) = c gives the family of integral curves (that is, the solutions) of the differential equation

Therefore, if a differential equation has the form

for some function f( x, y), then it is automatically of the form df = 0, so the general solution is immediately given by f( x, y) = c. In this case,

is called an exact differential, and the differential equation (*) is called an exact equation. To determine whether a given differential equation

is exact, use the Test for Exactness: A differential equation M dx + N dy = 0 is exact if and only if

Example 2: Is the following differential equation exact?

The function that multiplies the differential dx is denoted M( x, y), so M( x, y) = y 2 – 2 x; the function that multiplies the differential dy is denoted N( x, y), so N( x, y) = 2 xy + 1. Since

the Test for Exactness says that the given differential equation is indeed exact (since M y = N x ). This means that there exists a function f( x, y) such that

and once this function f is found, the general solution of the differential equation is simply

(where c is an arbitrary constant).

Once a differential equation M dx + N dy = 0 is determined to be exact, the only task remaining is to find the function f ( x, y) such that f x = M and f y = N. The method is simple: Integrate M with respect to x, integrate N with respect to y, and then “merge” the two resulting expressions to construct the desired function f.

___________________________

Additional information ❤

How do you find the derivative of y?

Implicit differentiation helps us find ​dy/dx even for relationships like that. This is done using the chain ​rule, and viewing y as an implicit function of x. For example, according to the chain rule, the derivative of y² would be 2y⋅(dy/dx).

____________________________

Hope it helps❤