Asked 7 years ago. Active 11 months ago. Viewed 83k times. Add a comment. Active Oldest Votes. Intuitive reasoning What does the derivative represent? Looking at different values of the absolute value function in some plots: Note that the tangent line is below the actual line for the absolute value function. Alice Ryhl Alice Ryhl 7, 2 2 gold badges 18 18 silver badges 40 40 bronze badges. 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. 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. 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. But we can also quickly see that the slope of the curve is different on the left as it is on the right. This suggests that the instantaneous rate of change is different at the vertex i. We use one-sided limits and our definition of derivative to determine whether or not the slope on the left and right sides are equal.
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.
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