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The function’s derivative is itself a function that is known as the second-order derivative. It is a differentiation process that can move along and continue till to get the third, fourth, and many successive derivatives of the function. And, this is known as the higher-order derivative. Below you will find some prominent examples of Higher-order derivatives related to the higher-order derivative and their solutions. It is very easy to get the higher-order derivative, but you must follow all the steps properly.
Meaning of Derivatives
The term derivative refers to the change in the quantity concerning the other thing. It helps get the moment and moment with the nature of an object. The derivative also includes some specific formulas. These formulas are as given below:
d/dx (k) = 0 (k = constant)
d/dx(x) = 1
d/dx(xn) = nxn-1
d/dx (kx) = k, (k is always same)
Types of Derivatives
There are many different types of derivatives. The details are given below:
1. First-order derivative:
The first-order derivative will also get from the tangent slope of the lane. It is a type of derivative that helps us tell us whether it’s an increasing function or a decreasing function. The rate of change specifically represents it.
2. Second-order derivative:
Another type of derivative helps us understand the graph shape given for the respective function. The function is divided according to the concavity. And further, the concavity is also divided into the main types, such as concave up and concave down.
Limits and Derivatives
When dx is made little to such an extent that it turns out to be barely anything, with Limits, we mean to say that x methodologies zero yet don’t become zero.
Numerically: for all genuine ε > 0 there exists a genuine δ > 0 to such an extent, for all x is 0 < |x − c| < δ and(here c ∈ R) we have |f(x) − L| < ε
Higher-Order Derivatives
The higher-order derivatives are the highest degree of derivatives that move along the differential equation process. Some of the best examples of Higher-order derivative are given below:
Example 1: Evaluate the main, second, and third derivative of f( x) = 5 x4 − 3×3 + 7×2 − 9x + 2.
Solution:
The first order derivative is 20×3-9×2+14x-9.
The second-order derivative is 60×2-18x+4.
The third-order derivative is 120x- 18.
Example 2: Evaluate f ′′ if f(x) = x cos x.
Solution:
It is given that f(x) = x cos x.
And f ′(x) = – x sin x + cos x,
Where,
f ′′(x) = – (x cos x + sin x) – sin x
= – x cos x – 2 sin x.
These examples of the higher-order derivatives will help you see how one can calculate the derivatives. The derivatives help us know about the rate of change of quantity concerning another.
What is the symbol of the derivative?
The symbol of the derivative is used to represent the derivative of a function, i.ef(x) is always equal to d/dx f(x) or with the derivative function, i.e., f’ (x). And, in case the function is denoted with the help of the variable y, the derivative first-order derivative & second-order derivative is specifically represented with y’ and y”. To reach the derivative function, it is a must that we have to apply the specific formula. And after its application, we will get the derivative of the function.
Why is the term called derivative?
This derivative term is assumed to be derived from one of the facts, i.e. the differential function, which is f′(x) and is designated by the main function f(x). So if (x) has been derived from another function, i.e. f(x), for each type of derivative, there is another function. And the same function in the starting will denote the function’s derivative.
Conclusion
From here, we see that there are different types of derivatives, whereas the term derivatives help denote the change in the quantity concerning the other quantity. The derivative is the specific term presented by the formula, i.e. dy and dx. We have to look at the specific formula at the time of calculation of the derivative. And if we will not apply the specific formula, then the function derivative will not get in the specific terms.
