What makes something not differentiable

These are the only kinds of non-differentiable behavior you will encounter for functions you can describe by a formula, and you probably will not encounter many of these. Now you have seen almost everything there is to say about differentiating functions of one variable. There is a little bit more; namely, what goes on when you want to find the derivative of functions defined using power series, or using the inverse operation to differentiating.

We will get to them later. We next want to study how to apply this, and then how to invert the operation of differentiation. As x approaches the corner from the left- and right-hand sides, the function approaches two distinct tangent lines. Since the function does not approach the same tangent line at the corner from the left- and right-hand sides, the function is not differentiable at that point.

The graph to the right illustrates a corner in a graph. Note: Although a function is not differentiable at a corner, it is still continuous at that point. Taking limits to find the derivative of a function can be very tedious and complicated.

The formulas listed below will make differentiating much easier. Each formula is expressed in the regular notation as well as Leibniz notation. Note: The graph of the derivative of a power function will be one degree lower than the graph of the original function. Note: For an example of a power function question, see Example 6 below. Constant Multiple Rule: If f is a differentiable function and c is a constant, then.

The chain rule is used to find the derivatives of compositions of functions. A composite function is a function that is composed of two other functions. For the two functions f and g, the composite function or the composition of f and g, is defined by. The function g x is substituted for x into the function f x. Often, a function can be written as a composition of several different combinations of functions. The chain rule allows us to find the derivative of composite functions.

The limits below are required for proving the derivatives of trigonometric functions. These limits and the derivatives of the trigonometric functions will be proven in your calculus lectures.

Here, they are simply stated. Note: These limits are used often when solving trigonometric limit problems. Try to remember them and the conditions under which they hold. Note: The derivatives of the co-functions cosine, cosecant and cotangent have a "-" sign at the beginning. This is a helpful way to remember the signs when taking the derivatives of trigonometric functions. The method of implicit differentiation allows us to find the derivative of an implicit function.

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. Calculus Derivatives Differentiable vs. Non-differentiable Functions. Jim H. Mar 13, There are three ways a function can be non-differentiable. We'll look at all 3 cases.

Case 1 A function in non-differentiable where it is discontinuous. Example 1d description : Piecewise-defined functions my have discontiuities.