Differentiation Solved Problems

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It's Here's a graph of the equation and the tangent line.

You can see that implicit differentiation allows us to do things with crazy, non-function equations that we might not have thought possible before.

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

The definition of derivative we used before involved a limit.

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.A function can be explicit or implicit: Explicit: "y = some function of x". Implicit: "some function of y and x equals something else". Free interactive tutorials that may be used to explore a new topic or as a complement to what have been studied already.$f(x)$ and $g(x)$ are differentiable functions, $C$ is real number: 1. Implicit Differentiation - Basic Idea and Examples What is implicit differentiation? Does the new definition of derivative involve limits as well?Do the rules of differentiation apply in this context?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.Our goal will be to find the derivative of the equation with respect to x, then find the zeros of its numerator (horizontal tangents) and the zeros of its denominator (vertical tangents).The function is If we take the implicit derivative, making sure to use the product rule on the second term (xy), we get: Solving for dy/dx gives us the derivative, a function of x and y. In this case, we can, however, solve the original equation for y: Dividing each term of the numerator by x gives us: We can take the derivative of y(x) directly now to get If we insert equation and simplifying the constant gives us a cubic equaiton that can be solved The result is the location of a single horizontal tangent at Vertical tangent(s) occur when the denominator of the derivative is zero.


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