When is x not differentiable

It allows us to differentiate y without solving the equation explicitly. We can simply differentiate both sides of the equation and then solve for y '. When differentiating a term with y , remember that y is a function of x.

The term is a composition of functions, so we use the chain rule to differentiate. For example, if you were to differentiate the term 3 y 4 it would become 12 y 3 y '. Note: For a more concrete demonstration of how to differentiate implicit functions, see example 14 below. Earlier in the derivatives tutorial, we saw that the derivative of a differentiable function is a function itself. If the derivative f' is differentiable, we can take the derivative of it as well. The new function, f'' is called the second derivative of f.

If we continue to take the derivative of a function, we can find several higher derivatives. In general, f n is called the nth derivative of f. Note: Recall that when working with motion application problems, the velocity of the particle is the first derivative of the displacement function.

The acceleration of the particle is the derivative of the velocity function, or equivalently, the second derivative of the displacement function.

It is often easy to calculate the exact value of a function at a point a , but rather difficult to compute values near a. We can find an approximate value of the function at points near a by using the tangent line to the curve at a. For more practice with the concepts covered in the derivatives tutorial, visit the Derivatives Problems page at the link below.

The solutions to the problems will be posted after the derivatives chapter is covered in your calculus course. To test your knowledge of derivatives, try taking the general derivative test on the iLrn website or the advanced derivative test at the link below. Please forward any questions, comments, or problems you have experienced with this website to Alex Karassev.

The derivative of a function f at a number a is denoted by f' a and is given by: So f' a represents the slope of the tangent line to the curve at a, or equivalently, the instantaneous rate of change of the function at a. The most common notations are : The notation is called Leibniz notation. The Derivative Function If we find the derivative for the variable x rather than a value a , we obtain a derivative function with respect to x.

Examples 3 Find the derivative of the function f at points A, B, C Differentiable Functions A function is differentiable at a if f' a exists. A function is not differentiable at a if its graph illustrates one of the following cases at a : Discontinuity A function is not differentiable at a if there is any type of discontinuity at a.

Vertical Tangent Line A function is not differentiable at a if its graph has a vertical tangent line at a. Corner A function is not differentiable at a if its graph has a corner or kink at a. Constant Multiple Rule: If f is a differentiable function and c is a constant, then Sum Rule: If f and g are differentiable functions, then In Leibniz notation, Note: For an example of the sum rule, see Example 7 below. Difference Rule: If f and g are differentiable functions, then In Leibniz notation, Note: For an example of the difference rule, see Example 8 below.

While the function is continuous, it is not differentiable because the derivative is not continuous everywhere, as seen in the graphs below. Get access to all the courses and over HD videos with your subscription. Get My Subscription Now. Please click here if you are not redirected within a few seconds. Home » Derivatives » Continuity and Differentiability. Have you ever wondered what makes a function differentiable?

Absolute Value — Piecewise Function. Then I used imagemagick to convert them into a gif. Thanks for the reference! A function can be differentiable while having a discontinuous derivative. I'd suggest googling discontinuous derivative for more info. If you want to see what's going on in your example, you can look into why a derivative can't have a jump discontinuity.

That is, if the derivative exists, and the limit of the derivative on both sides of the point exist, then these all must be equal. But the limit need not exist. Show 5 more comments. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.

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