Geometrically, the problem of finding the derivative of the function is existence of the unique tangent line at some point of the graph of the function. The problem of finding the unique tangent line at some point of the graph of the function is equivalent to finding the slope of the tangent line at the same point.
Let be a secant line trough points and . A secant line trough these two points has the slope:
If , that is, if point according to the graph of the function approaches point , then the secant line becomes the tangent line . Therefore, the slope of the secant line becomes the slope of the tangent line , that is
The expression is called the difference quotient for the function at point .
Let be an open interval. The function is differentiable at the point if exists
is called the derivative of the function at point and we write
The function is differentiable on if it is differentiable at every point .
Therefore, the equation of a tangent at point is
To find a tangent to the curve at point , it’s enough to find its slope, and that number is (provided that this limit exists).
If we let , then and as , the definition of the derivative of the function at point we can also write in the following form:
where is the change in -coordinates and is the change in -coordinates.
A derivative can also be written with symbols:
Let be an open interval and . If the function is differentiable on , then the function is continuous on .
The opposite is not true. A counterexample; the absolute value function defined as
is continuous on , however, it doesn’t have a derivative at point .
Let be an open interval and differentiable functions at point . Then
1.) the function is differentiable at point and
2.) (the Leibniz rule) the function is differentiable at point and
3.) the function , is differentiable at point and
4.) the function defined on is differentiable at point and
The derivative of a constant function , at is equal to
that is, .
The derivative of a function , at point is equal to
that is, .
For all is valid . For the potential we can write as a fraction:
where is a positive number.
That is, we have:
Example 1. Find the derivative of function if
Example 2. Find the derivative of function if
Example 3. Find the derivative of function if
Derivatives of trigonometric functions
Derivative of function sine.
We will use the trigonometric identity for difference of sines. Firstly, we calculate the difference quotient:
By using the definition of derivative we have:
Since , then
That is, we obtained .
Similarly, we can find derivatives of the rest trigonometric functions:
Higher order derivative
If the function has the derivative at every point of open interval , then the function also has the derivative on . The derivative of the function is denoted as and is called the second derivative of the function .
The th derivative of the function is denoted as and defined as
Example 4. Determine the third derivative of the function if:
The composite function rule
If and are differentiable functions, then a composite function is differentiable and the derivative of a composite function we calculate by the following formula:
The formula above is also known as the chain rule.
Example 5. Calculate the derivative of the following function:
Firstly, note that , where and .
Therefore, we have:
Now, by the chain rule, we have:
Derivative of inverse function
Let be injective function and . Then its inverse function is differentiable at point and
For instance, , that is, the function is an inverse function of function . By using the formula we calculate:
Derivative of logarithmic function
All logarithmic functions can be represented as logarithmic function of one base. Therefore, we choose , which base is the number .
Note. The function defined as is called the natural logarithm function.
The number is defined as the limit of a sequence:
Firstly, we calculate the difference quotient:
Now, by using the rule of logarithms we have:
By the definition of derivative now we have:
If we define then when . Since the domain of the function is , then , that is . Therefore, we can write . It follows that and , that is .
By the substitution we obtain:
By using the logarithm rule we have:
We defined , therefore
Derivative of logarithm of any other base we calculate from the connection of logarithm functions:
Derivative of exponential function
The exponential function is an inverse function of logarithmic function:
By using the rule for derivative of inverse function we have:
A derivative of the exponential function we calculate from the connection:
Example 6. Find the derivative of function if
Example 7. Calculate the first derivative of function at point if