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Figure %: The secant line connecting two points on the curve

To get the slope of the curve at the point (x, f (x)), let's now draw the tangent line at (x, f (x)).

Figure %: The tangent to the graph at (x, f (x)).

Recall that the tangent to the graph has the same slope as the graph at the point of tangency. Therefore, finding the slope of the graph at (x, f (x)) is the same as finding the slope of the tangent line we have just drawn.

Now comes a crucial step. Consider what happens to the secant line as h, the distance between the two points on the x-axis, is made progressively smaller:

Figure %: The distance separating the two points on the x-axis is made smaller

It appears now that as h gets smaller, the secant line looks more and more like the tangent line, which means that the slope of the secant is getting closer and closer to the slope of the tangent. This suggests that if we could make h arbitrarily small, the slope of the secant would get arbitrarily close to the slope of the tangent. Using limits, this idea could be represented as:

mtangent = (msecant)    


Substituting in the difference quotient for the slope of the secant yields

mtangent =    


Since the slope of the tangent is the same as the slope of the graph at the point of tangency, we can say:

slope off at(x, f (x)) =    


This is one of the central ideas of all of calculus. The limit of the difference quotient is such an important expression that it is given a name, the derivative, and is represented by "f'(x)". Thus, we can say:

f'(x) =    


is the derivative of the function f with respect to x.

The derivative gives the slope of the curve (also the slope of the tangent to the curve) at the point (x, f (x)). The derivative itself is also a function, because for every x value that it is given, it returns a value that is equal to the slope of the tangent to f at x.

An alternate notation for the derivative is the Leibniz Notation, when means "the derivative of whatever follows with the respect to x". Thus, means the derivative of f with respect to x, or f'(x) = means the derivative of y with respect to x. Since y commonly means f (x), this is usually the same as

f or f'(x)    

Differentiability

A function f is said to be differentiable at x = a if f'(a) exists. In other words, a function is differentiable at x = a if

   

exists.

Intuitively, for a function to be differentiable, it needs to be both continuous and "smooth". What is meant by "smooth" is that there are no sharp turns in the graph.

Tangent lines can only be drawn to graphs in places where they are both continuous and smooth, as shown below:

Figure %: A differentiable function with tangents drawn

One example of a function that is continuous but not "smooth" throughout is the absolute value function. Consider f (x) =|x|. This function is continuous, but has a sharp "corner" at x = 0:

Figure %: The graph of f (x) =|x| contains a corner.

The function f (x) =|x| is not differentiable at x = 0 because the sharp corner makes it impossible to draw a single tangent line, since there is no defined slope there. Thus, f'(0) does not exist for this function.

Differentiability Implies Continuity

Note that any differentiable function must also be continuous, since it is impossible to have a defined slope at a point of discontinuity. However, not all continuous functions are differentiable. An example of this was seen with the absolute value function.

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