Differentiation is arguably one of the most vital concepts in the world of Mathematics. The main reason is that it finds applications in a multitude of fields, both, directly and indirectly, related to Science. It is an indispensable part of the Calculus theory, with the other part being Integration. In this article you will learn about Implicit Differentiation.
The definition of Differentiation can be as the process of finding the derivative of a given function. A Function is an expression which relates one set of variables to another, where the two are interdepending.
In layman’s terms, differentiation is simply finding the instantaneous value of one set of the function with respect to another set. Sets usually consist of multiple variables.
Example of Implicit Differentiation
To make it more lucid, let us consider the following practical example:
When a car makes a journey from a source to a destination with varying velocity, the instantaneous velocity at any point of time during its journey can be found by differentiating the expression. The equation gives the car’s average velocity, with respect to the time taken during the journey.
Thus, the differentiation of the average velocity is with respect to the time. The result of this differentiation process is called the derivative. In the above example, the derivative is the acceleration of the car. It is the instantaneous change in velocity concerning time.
We have several formulas for differentiating a mathematical expression, the most fundamental among them being the formula for differentiating a single variable. If differentiation of a variable xn (read as x to the power of n) is with respect to x, the result is n*xn-1 (read as multiplication of n by x to the power of n minus one).
Similarly, when differentiation of an expression containing multiple variables is concerning another variable(which is to be present in the expression), evaluating the derivative of that expression.
This form of differentiation is explicit differentiation. It is such because of the sole reason that the differentiation of the expression is an explicit function of the variable concerning its differentiation.
As such, y = xn is an explicit function, the reason being that xn is an explicit function of y.
Comparison with Explicit Differentiation
Explicit means something obvious or stated clearly. In explicit differentiation, the expression is explicitly in relation to the variable with which differentiation of it takes place.
is the explicit differentiation of the relation y = xn.
Explicit differentiation is more or less straightforward in concern with the complexity of the expression. However, there is yet another method of differentiation, which is implicit. In this differentiation, the presentation of the expression is in an entirely different manner altogether, whose comparison to explicit functions takes place.
To illustrate the process of implicit differentiation, let us consider the following expression:
The above expression is an implicit function, as the variable for which the differentiation of expression is mixes within the whole expression.
To find the derivative of the above expression concerning, x, we should proceed as follows:
- Differentiate both sides of the equation for x:
- Rearrange the resultant equation to separate the dy/dx element from the rest of the equation by simple algebraic methods:
Therefore, in conclusion, that the implicit differentiation of the expression for x,
results in the derivative
In the example stated above, one may wonder, how differentiation of y2 for x takes place.
Expansion of the Differentiation Process
To understand this example further lets expand the differentiation process of such a term with the help of the Chain rule.
Let us consider y2 to be equal to u.
Now, according to the Chain rule:Such differentiation may seem like a much lengthier process compared to explicit differentiation. However, it has its advantages. You can use such differentiation to find the derivatives of expressions where you can’t perform explicit differentiation. A classic example of such a case is where there is more than one variable with varying powers. In problems related to explicit differentiation, we mostly encounter the differentiation of an expression containing only x(or a similar variable) with respect to x. Whereas, in the scenario of such differentiation, the expression may contain a multitude of variables with varying powers.
With a bit of practice and mastery of the Chain rule, one may easily solve problems of implicit differentiation. You can do this by skipping detailed calculation within each step, thereby making the process much more concise.