Derivatives: Definition, Rules, Examples, and Solution

Derivatives: Definition, Rules, Examples, and Solution

By: Answerout


The slope idea is often applied to straight lines. A straight line is defined as a function whose slope is constant. In other words, the inclination of a line remains constant regardless of whatever point we are looking at. The slope of a nonlinear function might change from one point to the next. As a result, we must introduce the concept of derivate, which allows us to determine the slope of these nonlinear functions at all places.

What is a Derivative?

The derivative of a function f at a point x, denoted by the symbol f ′(x), is given by: If such a limit exists. A function's derivative corresponds to the slope of its tangent line at a certain point on a graph. It is worth noting that the slope of the tangent line changes from point to point. The value of a function's derivative is thus determined by the moment at which we decide to evaluate it. We frequently refer to the slope of the function rather than the slope of its tangent line due to linguistic idiocy. The reverse function of derivative is known as antiderivative or integral. You can use an antiderivative calculator to find antiderivative of a function.
  • Derivatives of common functions

The following is a list of the most important derivatives. Although these formulas are formalizable, we shall merely mention them here. We recommend that you memorize them.

The constant function

Suppose f(x) to equal k, where k is a real constant. Following that, f′(x) = (k)′ = 0 Examples (8′) = 0 (-5)′ = 0 (0.2321)′ = 0

The identity function f(x) = x

Let f(x) = x be the x's identity function. Following that, (x) = (x)′ = 1 ′(x) = (x)′ = 1

xn function

Let f(x) = xn be a function of x with n being a real constant. We now have f′(x) = (xn)′ = n xn–1

Rules

  • The preceding rule applies to all sorts of exponents (natural, whole, fractional). It is critical, however, that this exponent remains constant. For exponential functions, another rule will need to be examined.
  • The identity function is a subset of the functions of type xn (with n = 1) and is derived in the same way: (x)′ = (x1)′ = 1 x1–1 = 1 x0 = 1 x1–1 = 1 x0 = 1
  • It is common for a function to satisfy this form yet require some reformulation before going to the derivative. It is the case of fractional exponents represented by roots (square, cubic, etc.).

The exponentiation function

Because both have exponents, it is relatively simple to mistake the exponential function with a function of the kind. They are, however, not the same. The exponent of an exponential function is a variable. Let f(x) = (ax)′ = ax ln (a)
  • Fundamental Derivation Principles

In general, we will have to deal with not just the functions mentioned above, but also their combinations: multiples, sums, products, quotients, and composite functions. As a result, we must give the rules that enable us to deduce these more complicated scenarios.

Constant multiples

Suppose k to be a real constant and f(x) to be any specified function. Following that, (K f(x)) ′ = k f′ (x) In other words, we may ignore the constant that remains constant and merely deduce the function of x.

Function addition and subtraction

Consider the functions f(x) and g(x). Following that, (F(x) ± g(x))′ = (f′ (x) ± g′ (x)) When we derive a sum or a subtraction of two functions, the preceding rule indicates that the functions may be derived separately without modifying the operation that connects them.

Product Rule

Consider the functions f(x) and g(x). Then the product's derivate will be, (f(x) g(x))′ = f′ (x) g(x) + f(x) g′ (x) We must strictly adhere to this principle and resist the impulse to write. (f(x) g(x))′ = f′ (x) g′ (x) Which is an incorrect assertion.
  • Composite Function Derivation

A composite function is one that has the form f (g(x)). How can we tell whether a function is composite? A composite function is a function that includes another function. A composite function is one that can be broken down into many components, each of which is a function in and of itself, and these pieces are not related by addition, subtraction, product, or division. A composite function is, for example, the function f(x) = ex^3. We may format it as follows: f(g(x)) where g(x) = x3. In contrast to the non-composite function f(x) = x3ex. It is just the result of functions. Here are a composite function example:
  • f(x) = ln(x2 + 2x + 1)
This function may be written as f(g(x)) = ln (g(x)), where g(x) = x2 + 2x + 1.
  • f(x) = e4x-2
This function may be written as f(g(x)) = eg(x), where g(x) = 3x - 5.
  • f(x) = ln(x) + 3x – ex
This function may be written as f(g(x)) = (g(x))4, where g(x) = ln(x) + 3x – ex

The chain rule

Consider f and g to be two functions. The derivative of the composite function f(g(x)) is then computed as, f(g(x)) ′ = f′ (g(x)) g′ (x) Alternatively, f((u))′ = f′(u) u′, where u = g(x) To evaluate chain rule using an online tool, you can use derivative calculator which can assist you every type of derivative calculations. According to the chain rule, in order to derive a composite function, we must first deduce the exterior function (the one that contains all others) while leaving the interior function alone and then multiplying it by the internal function's derivative. The method is repeated if the latter is also composite. Keep an eye out because the internal function might also be a product or a quotient, etc.
  • Calculation of the slope of a tangent at a single point

As stated before, the derivative function f′(x) reflects the slope of the tangent line at f(x) at all x points. We will frequently be required to evaluate this slope at a certain location on the graph. To compute the slope of the tangent of the function f(x) when x = 1, for example, we cannot compute f(1) and deduce this number. We would then get a slope of 0 because f(1) is a constant. Instead, we must calculate the derivative f′(x) at all places and then evaluate it at x = 1. To express the derivative of the function evaluated at the point x = a, we shall use the notation f′(a).
  • Functions for increasing and decreasing values

There is a direct link between a function's growth and decrease and the value of its derivative at one moment.
  • If the derivative is negative at a particular point, it means that the function is declining at that moment.
  • If the derivative is positive at a particular point, it means that the function is rising at that moment.

Summary

In this post, we have described the derivatives along with the rules that you may encounter at some point in your studies. Derivatives have huge applications in mathematics especially in calculus and it is not possible to cover each one of them in one post. We will cover more about this topic in the next posts. Till then, you can practice the listed rules with several functions to master them. All of the rules explained above should be implemented accordingly. For example, the product rule can only be applied if a function contains a product of values or variables. Use this differentiation calculator to facilitate the derivations if you are still confused about solving derivatives.

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