What does not differentiable mean

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. A function is formally considered differentiable if its derivative exists at each point in its domain, but what does this mean?

So, as long as you can evaluate the derivative at every point on the curve, the function is differentiable. Hence, differentiability is when the slope of the tangent line equals the limit of the function at a given point. This directly suggests that for a function to be differentiable, it must be continuous, and its derivative must be continuous as well.

Now, this leads us to some very important implications — all differentiable functions must therefore be continuous, but not all continuous functions are differentiable! If there was a hole in the line at 2,3 and there is another point at 2,1 , then would the graph be differentiable at that point and why? Question Question c5. Question 0b8db. Question b1. Question e8. Integrating factor question? Derivatives View all chapters. Figure: na For functions of more than one variable, differentiability at a point is not equivalent to the existence of the partial derivatives at the point; there are examples of non-differentiable functions that have partial derivatives.

Comments S. References [a1] E. Hewitt, K. Stromberg, "Real and abstract analysis" , Springer [a2] K. Stromberg, "Introduction to classical real analysis" , Wadsworth How to Cite This Entry: Non-differentiable function.