# 6.1 Implicit Vs Explicitap Calculus

6.1worksheetans.pdf: File Size: 196 kb: File Type: Download File. Proudly powered by Weebly. Differentiating Explicit and Implicit Functions. An explicit function is one which is given in terms of the independent variable. Take the following function, y = x 2 + 3x - 8 y is the dependent variable and is given in terms of the independent variable x. Note that y is the subject of the formula. Please assume this is my initial exposure to calculus and derivatives. I am having difficulty making the connection between the application of the chain rule to explicit differentiation and that of implicit differentiation. Everything I’ve learned so far about differentiation has been based on explicitly defined functions and limits. Let's try now to use implicit differentiation on our original equality to see if it works out: We must use the product rule again in the left side: Now we must substitute y as a function of x to compare it to our first result: And we got the same result, as expected. Return to Implicit Differentiation. Not all implicit equations can be restated explicitly in a single equation. For example, the implicit equation x 2 +y 2 = 9 needs two explicit equations, which are the top and bottom halves of a cricle respectively, to define the functional relation completely.

Anexplicit function is one which is given in terms of
the independent variable.

Take the following function,

y = x2 + 3x - 8

y is the dependent variable and is given in terms of the
independent variable x.
Note that y is the subject of the formula.

Implicit functions, on the other hand, are usually given in terms
of both dependent and independent variables.

eg:- y + x2 - 3x + 8 = 0 Sometimes, it is not convenient to express a function explicitly.
For example, the circle x2 + y2 = 16 could be written as

or

Which version should be taken if the function is to be
differentiated ?

It is often easier to differentiate an implicit function without
having to rearrange it, by differentiating each term in turn.
Since y is a function of x, the chain, product
and quotient rules apply !

Example

Differentiate x2 + y2 = 16 with respect to x.

Compared to Example

## 6.1 Implicit Vs Explicitap Calculus Calculator

Differentiate 2x2 + 2xy + 2y2 = 16 with respect to x.

Example

Find the gradient of the tangent at the point R(1,2)
on the graph of the curve defined by x3+ y2= 5, and determine
whether the curve is concave up or concave down at this point.

## 6.1 Implicit Vs Explicitap Calculus Algebra

Divide through by y

Now substitute to find the particular solution