Theorems On Differentiation

Differentiation formulas are formulas that you can use for different functions and then find the derivative of the resulting function. There are many different differentiation formulas, some more useful than others. In this article, we’ll take a look at the most useful differentiation formulas and show you when they come in handy in day-to-day calculations. We’ll also talk about how to use these formulas to differentiate any function by hand.

Before you can figure out how to apply differentiation formulas to your problems, you need to know what the most important ones are and how they’re used. The two most widely used differentiation formulas are the Power Rule and the Product Rule, which we’ll discuss in this article. For related reading, check out our write-up on How To Differentiate Integrals By Using Integration Formulas.

When u = 0, we get the Mean Value Theorem

The Mean Value Theorem states that if we have a continuously differentiable function, we can find the average value of that function by taking the average of the value at the endpoints: f(x) = x2 between 0 and 2, then 3x – 6 is a continuously differentiable function, so using the Mean Value Theorem we get: Average Value = (0 + 2)/2 = 1.5.

When u = 1, we get the Power Rule

The Power Rule says that the derivative of u raised to the power k is equal to Ku times k, but with a caveat: if u is negative, we must flip the sign. That’s it! In other words, when x = 1 we get y = x2, but when y = -1 we get x = -x2. Easy peasy! That’s all there is to the Power Rule.

When u = π/2, we get the Product Rule

Most differentiation formulas have the answer u = π/2. So, before moving on to understand how we differentiate in the first place, you need to learn the Product Rule: u = u. d. d. This can be derived from the Integral Rule through some slight modifications. From calculus, we know that du/dx = u-v (Integral Rule). If we treat x as a constant in Integration and differentiate it separately, then du = dx/x and hence du/du = dx/dx.

When u = 0 and v = 0, we get the Quotient Rule

We get the Quotient Rule when u = 0 and v = 0, or equivalently when the expression inside the radical sign is 0. That’s all there is to it. Now, let’s look at some other simple rules involving differentiation formulas.

When both u and v are greater than 0 but less than 1, we get the Chain Rule

One of our first important differentiation formulas is also one of our most complex. But fear not! We’ll break it down into simple-to-understand pieces and show you just how much value it has in other situations. The chain rule, as its name implies, describes what happens when you take a derivative concerning more than one variable at once.

For More Advanced Students: Lagrange Multipliers

In that section, we focused on how differentiation affected a single function, not a system of functions. In that section, we gave a quick example of creating a linear system and then differentiated it to show how to solve it. But when you are faced with a more complex problem with multiple variables or complicated multiple steps in your thinking process, you’ll have to use mathematical tools such as Lagrange multipliers to help solve your system.

Conclusion

Differentiation formulas are used to simplify and derive the derivative of an algebraic function, given that you know the derivative of every term in the expression. The most common derivative formulas are used in mathematics, science, engineering, and economics to calculate derivatives of polynomials, trigonometric functions, exponential functions, logarithmic functions, and their composite functions. Differentiation formulas are also used in calculus to derive more complex differential equations from simple ones.