Two Derivative Concept You Must Know

Implicit differentiation Derivative:

Implicit differentiation is a method of finding the derivative of an implicitly defined function. In implicit differentiation, we differentiate both sides of an equation with respect to one variable, and then we solve for the derivative of the other variable. If you want to find Implicit differentiation derivative quickly without any manual calculation, you can use the differentiation calculator for accurate results and solution.

Example:

 Consider the equation x^2 + y^2 = 1. In this equation, y is implicitly defined as a function of x. To find the derivative of y with respect to x, we can use implicit differentiation. We take the derivative of both sides of the equation with respect to x.

dx/dt(x^2 + y^2) = dx/dt(1)

2x + dy/dx*dy/dt(x) = 0

then we can solve for dy/dx by dividing both sides by 2x.

dy/dx = -x/(y)

Another example is y^3 = x^2y+ x^3

we can differentiate both sides with respect to x,

3y^2dy/dx = x^2dy/dx + 2xy +3x^2

then we can solve for dy/dx by dividing both sides by x^2 + 2y

dy/dx = x/(x^2 + 2y)

Must know: 2 important Integral Concepts in Calculus

Partial Derivative:

Partial derivative is a concept in calculus that deals with the rate of change of a function with respect to one variable, while keeping the other variables constant. It is used to find the rate of change of a multi-variable function with respect to one of the variables. It is represented by the symbol ∂/∂x (or ∂/∂y, ∂/∂z, etc.) where x, y, z are the variables. You can also calculate the derivative of a curve with numerous variables online by using partial differentiation calculator.

Example:

Consider the function f(x,y) = x^2 + y^2. To find the partial derivative of f with respect to x, we need to keep y constant and find the rate of change of f with respect to x.

∂/∂x (x^2 + y^2) = ∂/∂x (x^2) = 2x

To find the partial derivative of f with respect to y, we need to keep x constant and find the rate of change of f with respect to y.

∂/∂y (x^2 + y^2) = ∂/∂y (y^2) = 2y

Another example is the function z = f(x,y) = x^2*y^3. To find the partial derivative of f with respect to x,

∂/∂x (x^2*y^3) = ∂/∂x (x^2) * y^3 + x^2 * ∂/∂x (y^3) = 2xy^3

To find the partial derivative of f with respect to y,

∂/∂y (x^2y^3) = x^2 * ∂/∂y (y^3) = 3x^2y^2

Conclusion:

Partial derivative is a concept in calculus that deals with the rate of change of a multi-variable function with respect to one of the variables, while keeping the other variables constant. It's represented by the symbol ∂/∂x (or ∂/∂y, ∂/∂z, etc.) and it can be found by differentiating one variable at a time and keeping the other variables constant. Partial derivatives are useful in physics, engineering, and other fields to find the rate of change of a multi-variable function with respect to one of the variables.

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Implicit differentiation is a method of finding the derivative of an implicitly defined function by differentiating both sides of the equation with respect to one variable and solving for the derivative of the other variable. Implicit differentiation can be useful when the equation is not written in an explicit form, and it is not possible to solve for one variable in terms of the other.

 

Author Bio: I am Amelia Margaret, Lecturer at University of Alabama & having PhD in Math’s.

LinkedIn: linkedin.com/in/amelia-margaret-104365262