How do I find the derivative of a function?

The Derivative of Function f The derivative of a function f is an expression that tells you what the slope of f is at any point in the domain of f. The derivative of f is a function itself. In this article, we will focus on the 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.

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, you 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; 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, the function is differentiable; if it does not, then the function is not differentiable. If you are unfamiliar with limits or want to know more about them, you might want to read up on 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. If it exists, then you have the derivative, or else you know the function is not differentiable. Example 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 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) Properties of the derivative Properties of the derivative Known Derivatives 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. Polynomials 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. Goniometric Functions Of course, the sine, cosine and tangent also have a derivative. They are pretty easy to calculate if you know the standard rule. These rules 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. Math: How to Find the Tangent Line of a Function in a 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. Math: How to Find 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.

assuming it’s a linear function, such as f(x) = x^3 you simply times by the power and take one away from it you’d get f’(x) = 3x^2

In simple terms, there's a table of derivatives you can learn, or you can look for the limit when h tends to 0 of f(x+h)-f(x)/h. If you're asked for a proof, you have to calculate the limit. Else, just practice a lot using the table, and eventually it'll come naturally. Here's a link - https://www.adamponting.com/differentiation-rules-and-standard-derivatives/.

Hi Rachel, To find the derivative of a function it depends on the type of function here is a few examples of different types. If y = 3x^3 the dy/dx=3*3x^2=9x^2 Notice how we drop the power write the x and then take 1 from the power. If y=4x^2+2x+3 dy/dx = 8x +2 Notice how if the power of x is 1 we just remove the x and notice that the derivative of a constant is 0. If you have a function in the form (ax+b)^c where a, and c are constants. Then we drop the power, write out the bracket and subtract one from the power and then we also multiply by the derivative of the bracket. For example If y=(5x-1)^4 then dy/dx=(4(5x-1)^3)*5 dy/dx=20(5x-1)^3 Notice how what it is in the bracket doesn’t change. If you have a trigonometric function then the derivative of sin(x) = cos(x) and the derivative of cos(x) = -sin(x) if there is a number in front of the x then we multiply by the number in front of the x as well. If y=sin(3x) then dy/dx = 3sin(3x). Notice how the function inside the bracket again doesn’t change. Kind regards Owen

The general formula for deriving a function is: if f(x)=x^n, then f'(x)=nx^(n-1). For example: f(x)=x^4 f'(x)=4x^3. I hope this helps :).