I studied applied mathematics, in which I did both a bachelor's and a master's degree.
The derivative of a function f is an expression that tells you what the slope of f is in any point in the domain of f. The derivative of f is a function itself. In this article, we will focus on functions of one variable, which we will call x. However, when there are more variables, it works exactly the same. You can only take the derivative of a function with respect to one variable, so then you have to treat the other variable(s) as a constant.
Definition of the Derivative
The derivative of f(x) is mostly denoted by f'(x) or df/dx, and it is defined as follows:
f'(x) = lim (f(x+h) - f(x))/h
With the limit being the limit for h goes to 0.
Finding the derivative of a function is called differentiation. Basically, what you do is calculate the slope of the line that goes through f at the points x and x+h. Because we take the limit for h to 0, these points will lie infinitesimally close together; and therefore, it is the slope of the function in the point x. Important to note is that this limit does not necessarily exist. If it does, then the function is differentiable; and if it does not, then the function is not differentiable.
If you are not familiar with limits, or if you want to know more about it, you might want to read my article about how to calculate the limit of a function.
How to Calculate the Derivative of a Function
The first way of calculating the derivative of a function is by simply calculating the limit that is stated above in the definition. If it exists, then you have the derivative, or else you know the function is not differentiable.
As a function, we take f(x) = x2.
(f(x+h)-f(x))/h = ((x+h)2 - x2)/h
= (x2 + 2xh +h2 - x2)/h
= 2x + h
Now we have to take the limit for h to 0 to see:
f'(x) = 2x
For this example, this is not so difficult. But when functions get more complicated, it becomes a challenge to compute the derivative of the function. Therefore, in practice, people use known expressions for derivatives of certain functions and use the properties of the derivative.
Properties of the Derivative
Calculating the derivative of a function can become much easier if you use certain properties.
- Sum rule: (af(x)+bg(x))' = af'(x) + bg'(x)
- Product rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
- Quotient rule: (f(x)/g(x))' = (f'(x)g - f(x)g'(x))/g(x)2
- Chain rule: f(g(x))' = f'(g(x))g'(x)
There are a lot of functions of which the derivative can be determined by a rule. Then you do not have to use the limit definition anymore to find it, which makes computations a lot easier. All these rules can be derived from the definition of the derivative, but the computations can sometimes be difficult and extensive. Knowing these rules will make your life a lot easier when you are calculating derivatives.
A polynomial is a function of the form a1 xn + a2xn-1 + a3 xn-2 + ... + anx + an+1.
So a polynomial is a sum of multiple terms of the form axc. Therefore by the sum rule if we now the derivative of every term we can just add them up to get the derivative of the polynomial.
This case is a known case and we have that:
d/dx xc = cxc-1
Then the derivative of a polynomial will be:
na1 xn-1 + (n-1)a2xn-2 + (n-2)a3 xn-3 + ... + an
Negative and Fractional Powers
d/dx xc = cxc-1 does also hold when c is a negative number and therefore for example:
1/x = x-1
d/dx 1/x = -1/x2
Furthermore, it also holds when c is fractional. This allows us to calculate the derivative of for example the square root:
d/dx sqrt(x) = d/dx x1/2 = 1/2 x-1/2 = 1/2sqrt(x)
Exponentials and Logarithms
The exponential function ex has the property that its derivative is equal to the function itself. Therefore:
d/dx ex = ex
Finding the derivative of other powers of e can than be done by using the chain rule. For example e2x^2 is a function of the form f(g(x)) where f(x) = ex and g(x) = 2x2. The derivative following the chain rule then becomes 4x e2x^2.
If the base of the exponential function is not e but another number a the derivative is different.
d/dx ax = ax ln(a)
where ln(a) is the natural logarithm of a.
The derivative of the logarithm 1/x in case of the natural logarithm and 1/(x ln(a)) in case the logarithm has base a.
Of course the sine, cosine and tangent also have a derivative. They are pretty easy to calculate if you know the standard rule. These rule are again derived from the definition but they are not so obvious. You need Taylor expansions to prove these rules, which I will not go into in this article. Instead I will just give the rules.
d/dx sin(x) = cos(x)
d/dx cos(x) = - sin(x)
d/dx tan(x) = 1 - tan2(x)
d/dx arcsin(x) = 1/sqrt(1-x2)
d/dx arccos(x) = -1/sqrt(1-x2)
d/dx arctan(x) = 1/(1+x2)
Applications of the Derivative
The derivative comes up in a lot of mathematical problems. An example is finding the tangent line to a function in a specific point. To get the slope of this line, you will need the derivative to find the slope of the function in that point.
Another application is finding extreme values of a function, so the (local) minimum or maximum of a function. Since in the minimum the function is at it lowest point, the slope goes from negative to positive. Therefore, the derivative is equal to zero in the minimum and vice versa: it is also zero in the maximum. Finding the minimum or maximum of a function comes up a lot in many optimization problems. For more information about this you can check my article about finding the minimum and maximum of a function.
Furthermore, a lot of physical phenomena are described by differential equations. These equations have derivatives and sometimes higher order derivatives (derivatives of derivatives) in them. Solving these equations teaches us a lot about, for example, fluid and gas dynamics.
Multiple Applications in Math and Physics
The derivative is a function that gives the slope of a function in any point of the domain. It can be calculated using the formal definition, but most times it is much easier to use the standard rules and known derivatives to find the derivative of the function you have.
Derivatives have a lot of applications in math, physics and other exact sciences.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.
Umesh Chandra Bhatt from Kharghar, Navi Mumbai, India on November 30, 2020:
Mathematics was my favourite subject till my graduation. Yoy have explained the derivative nicely. Thanks.