# Unraveling the Implicit Differentiation of y=vx

• thomas49th
In summary, implicit differentiation is used when both variables in an equation are functions of the same independent variable and cannot be treated as constants.
thomas49th

## Homework Statement

Consider
y = 2a + ax

find dy/dx

dy/dx = a

That is right is it not, as a is treated merly as a constantNow consider this question:

Use the substitution y = vx to transform the equation:

dy/dx = (4x+y)(x+y)/x²

into

x(dv/dx) = (2+v)²

According to the mark scheme they
differentiate dy/dx implicitally
y = vx
dy/dx = x(dv/dx) + v

BUT why have we differentitated implicitally?

Thanks :)

In your first example, a is assumed to be a constant, so dy/dx = a, as you showed.
In your second example, both x and y are variables, and v is some function of x. In the substitution y = vx, when you differentiate the right side with respect to x, you cannot treat v as a constant as you did in the first example, so you have to use the product rule. You are assuming that both y and v are functions of x, so any differentiation has to be done implicitly.

## What is implicit differentiation and how is it related to y=vx?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly written in terms of its independent variable. In the equation y=vx, implicit differentiation is used to find the derivative of y with respect to x.

## What are the steps to perform implicit differentiation on y=vx?

The steps to perform implicit differentiation on y=vx are as follows:

1. Write the equation in the form of f(x,y)=0.
2. Differentiate both sides of the equation with respect to x.
3. For each y term, multiply by dy/dx.
4. Collect all terms with dy/dx on one side and factor it out.
5. Solve for dy/dx to find the derivative of y with respect to x.

## What is the importance of implicit differentiation in mathematics?

Implicit differentiation is important in mathematics because it allows us to find the derivative of a function even when it is not explicitly written in terms of its independent variable. This is useful in solving many real-world problems in physics, engineering, and economics, where relationships between variables are often not explicitly stated.

## What is the difference between implicit differentiation and explicit differentiation?

Explicit differentiation is used to find the derivative of a function that is explicitly written in terms of its independent variable, while implicit differentiation is used to find the derivative of a function that is not explicitly written in terms of its independent variable. In other words, explicit differentiation can be used when the function is in the form of y=f(x), while implicit differentiation is used when the function is in the form of f(x,y)=0.

## How can implicit differentiation be applied in real-world scenarios?

Implicit differentiation can be applied in many real-world scenarios, such as finding the rate of change in physics problems involving motion, determining the optimal production level in economics, or calculating the growth rate of a population in biology. It is a useful tool for analyzing relationships between variables and making predictions in various fields of study.

Replies
2
Views
1K
Replies
3
Views
1K
Replies
5
Views
1K
Replies
2
Views
1K
Replies
6
Views
3K
Replies
5
Views
2K
Replies
6
Views
1K
Replies
32
Views
3K
Replies
20
Views
2K
Replies
1
Views
1K