Differentiation Solved Problems

Differentiation Solved Problems-41
Implicit differentiation is really just application of the chain rule, where we recognize y as a function of x, and further differentiate any term containing y using the chain rule.For example, It's possible to solve for y in this equation, of course, and then find dy/dx, but implicit differentiation makes finding the derivative much easier.Then the equation of the tangent is easy to find from the point and the slope.

Taking the implicit derivative, we see that it goes to zero when x goes to zero: Here is a graph of the function showing the single horizontal tangent and the vertical tangent, at which the function has no value.

When dealing with a function of more than one independent variable, several questions naturally arise.

The basic idea about using implicit differentiation 1.

Implicit differentiation is a very powerful technique in differential calculus.

It allows us to find derivatives when presented with equations and functions like those in the box.

→ One could solve for y and find y'(x), but there's an easier way, and it applies to the derivatives of more complicated functions, too.

We start by taking the derivative with respect to x (we could as easily take it with respect to y) of each term on both sides.

We apply the sum rule (the derivative of a sum is the sum of derivatives) on the left and recall that the derivative of a constant is zero.

For example, how do we calculate limits of functions of more than one variable?

The definition of derivative we used before involved a limit.


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