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How do We Understand Implicit Differentiation and Partial Derivatives?


We’ve already looked at how to find tangent line equations for functions and the rate of change of a function at a given position. We had the exact equation for the function in all circumstances, and we explicitly take differentiation of these functions.

Think about the case where we want to find the rate of change of an arbitrary curve or the equation of a tangent line to an arbitrary curve at a specific point. We solve these questions by shaping the derivatives of functions that describe in terms of implicit.

Implicit Differentiation

When the dependent variable is a function of the independent variable, we represent it in terms of explicit function. When we write the equation, for example, we are explicitly defining it. On the other hand, the implicit function-variable relation is defined by an equation not in the form of terms. The function is implicitly defined in the equation, for example.

To get the slopes of tangents to curves, we can utilize implicit differentiation, which isn’t working (they don’t pass the vertical line test). 

How to Solve Implicit Differentiation?

The general pattern to solve implicit differentiation is:
  1. We will start with the inverse function in explicit form. e.g y = sin1(x)
  2. Then we convert in non-inverse format. i.e. x = sin(y)
  3. Now, we differentiate this function with respect to x on both sides of the equation.
  4. In the last step, we will solve for dy/dx.

But besides that manual method, we can also calculate implicit differentiation of implicit function by using an online tool like an implicit derivative calculator with steps.

Applications of Implicit derivative:

The main application of implicit differentiation which differs them from other techniques of derivatives are as follow:

  • To check the temperature variation.
  • It allows computing the derivative of an inverse function.

How to differentiate implicitly?

But now learn how to calculate the derivative:

  • Calculate each variable’s derivative.
  • Multiply by dy/dx whenever you compute the derivative of “y.”
  • Solve the dy/dx remaining equation 

Partial derivative:

In general, if a function satisfies an equation, it is considered implicitly defined.

In differential calculus, a Partial derivative is the derivative of a function of numerous variables with regard to a change in only one of its variables. Partial derivatives and partial differential equations are useful in examining surfaces for maximum and lowest points. A first partial derivative, like ordinary derivatives, denotes the rate of change or slope of a tangent line. Two first partial derivatives present the slope in each of two perpendicular ways for a three-dimensional surface. The partial derivatives of the second, third, and higher orders provide more information about how the function changes at any given position.

How to Solve Partial Derivative?

The partial differentiation is as simple to solve as a simple problem for differentiation. We simply treat the “y” variable as a constant number. And then, we solve the whole question problem with respect to “x” or vice versa.

But with the help of a partial differentiation calculator with steps, we can calculate the partial derivative of a function with just a single click.

Applications of Partial Derivative

There are a lot of different real-life applications of partial derivatives. The main application of partial differentiation which differs them from other techniques of derivatives are as follow:

  • Marginal functions
  • Marginal rate of substitution (MRS)
  • Measuring the slope
  • Finding the extreme values


Only the variable with respect to which the function is being differentiated is considered a variable in partial differentiation, while all other variables are treated as constants. All differentiated terms variables will consider variables in ordinary differentiation.

With implicit differentiation, both variables are differentiated, but at the end of the solution, one variable is separated (without any integer being associated with it) on one side. On the other hand, Partial differentiation differentiates one variable while keeping the other constant.

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