The Concept of the Derivative#
The Basic Definitions#
Slope of a Tangent Line
The slope of the tangent line to the graph of \(f(x)\) at the point \(P(x, f(x))\) is given by
if the limit exists.
Average and Instantaneous Rate of Change
The average rate of change of \(f(x)\) over the interval \((x, x+h)\) or slope of the secant line to the graph of \(f(x)\) through the points \((x,f(x))\) and \((x+h, f(x+h))\) is
The instantaneous rate of change of \(f(x)\) at \(x\) or slope of the tangent line to the graph of \(f(x)\) at \((x,f(x))\) is
if the limit exists.
The Limit Definition of the Derivative
The derivative of a function \(f(x)\) with respect to \(x\) is the function \(f'(x)\) which is defined as follows
and the domain of \(f'(x)\) is the set of all \(x\) where the limit exists.
Differentiability and Continuity
If a function is differentiable at \(x=a\), then it is continuous at \(x=a\). If a function is continuous at \(x=a\), then it is not necessarily differentiable at \(x=a\).
Computing Derivatives Using the Limit Definition#
Example 1#
Find the slope of the tangent line to the function
at \(x=5\) using the limit definition of the derivative.
Step 1: Write down the limit definition of a derivative.
Remember: \(f(x+h)\) means that we take \(f(x)\) and replace \(x\) with \((x+h)\). For example, if \(f(x) = x^2\), then \(f(x+h)=(x+h)^2=x^2+2xh+h^2.\)
Step 2: Plug \(f(x+h)\) and \(f(x)\) into definition.
Plug \(f(x+h)\) and \(f(x)\) into the limit definition of the derivative. Using brackets will help avoid errors from forgetting to distribute the negative sign:
Step 3: FOIL and Simplify
Step 4: Factor out \(h\) and cancel.
Factor out an \(h\) in the numerator and cancel it with the factor of \(h\) in the denominator.
Step 5: Evaluate the limit.
Step 6: Plug \(x=5\) into evaluated limit.
We have found that \(f'(x) = 6x\) is the derivative of our function and the general form of the slope of the tangent line. All that’s left for us to do is to plug in \(x=5\). Therefore, the slope of the tangent line when \(x=5\) is \(30\).
Example 2#
Find the derivative of the function
using the limit definition of the derivative.
Step 1: Write down the limit definition of a derivative.
Step 2: Plug \(f(x+h)\) and \(f(x)\) into definition.
Plug \(f(x+h)\) and \(f(x)\) into the limit definition of the derivative.
Step 3: Rationalize, FOIL, and Simplify.
Notice how making use of the formula \((\sqrt{A} - \sqrt{B})(\sqrt{A} + \sqrt{B}) = A - B\) can help eliminate some of the above computations.