Derivative Examples

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In this article, we are going to discuss derivative questions and answers, a few solved derivative problems. Let’s first discuss what derivatives are.

Derivatives are defined as the varying rate of change of a function with respect to an independent variable. The derivative is mainly used when a quantity varies, and the rate of change is not constant. The derivative is generally used to measure the sensitivity of one variable (dependent variable) with respect to another variable (independent variable). Here, we are going to discuss what derivatives are and the definition of derivatives Mathematics, limits, and derivatives in detail.


Derivatives Meaning

Derivatives Mathematics points to the instantaneous rate of change of a quantity with respect to the other quantity. It is very helpful to analyze the nature of a given amount.


Derivative Definition

A function that denotes the rate of change of the other function can be called the derivative of that function. The method of finding a derivative can be called differentiation. It is represented as a dependent variable in terms of an independent variable through an equation. The derivative is a value that changes with respect to its input. A derivative is denoted as f’(x).


Derivative Examples

Let’s discuss a few derivative examples. Consider a function that involves the change in velocity of a vehicle moving from one point to another. The change in velocity is certainly dependent on the speed and direction in which the vehicle is travelling. If the acceleration is to be calculated, then the limits of the function are essential. The theory of derivative is derived from limits. In this write-up, we have put together the concepts of derivatives along with a few solved derivative examples.

Let a car take ‘t’  seconds to move from a point ‘a’ to ’b’.

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But how long will it take to move from point ‘a’ to ‘c’?

Or

How much distance will the car cover in ‘t-1’ seconds?

This can be known from the velocity, as follows.

Velocity (v) = d(x)/d(t)

where ‘x’ is the distance travelled and ‘t’ is the time taken to cover that distance.

This will give us the distance covered per unit time which will help us to analyze any distance covered in any interval of time.


Derivatives Math – Calculus

The process of finding the derivatives is known as differentiation. The inverse process is called anti-differentiation. Let us consider the derivative of a function to be y = f(x).  It can be defined as the measure of the rate at which the value of y changes with respect to the change of the variable x. It can also be known as the derivative of the function namely “f”, with respect to the variable x.

If an infinitesimal change in x can be denoted as dx, then the derivative of y with respect to x can be written as dy/dx.

So here, the derivative of y with respect to x is shown as “dy by dx” or “dy over dx”

Example:

Let there be ‘y’ which is a dependent variable and ‘x’ be an independent variable.

Let there be a change in the value of x, which is dx.

The changes in x will mostly bring up a change in y, let that be dy.

Now to find out the change in y with a unit change in x we can follow the steps given below.

Let there be a function namely f(x) whose value varies as the value of x varies

Steps to find the Derivative:

  1. First, you need to change x by the smallest possible value and let that be ‘h’ and so the function becomes f(x+h).

  2. Now you need to get the change in the value of the function that is: f(x + h) – f(x).

  3. The rate of change in function f(x) on changing from ‘x’ to ‘x+h’ will be equal to \[\frac{dy}{dx}\] = \[\frac{f(x+h) - f(x)}{h}\].

Now d(x) is negligible because it is considered to be too small.


First-Order Derivative

The first-order derivatives basically tell us about the direction of the function whether the function is increasing or decreasing. The first derivative mathematics or the first-order derivative can also be said to be an instantaneous rate of change. This can also be predicted from the slope of the tangent line.


Second-Order Derivative

The second-order derivatives are generally used to get an idea of the shape of the graph for the given function. We can define functions in terms of concavity. The concavity of the given graph function can be classified into two types namely:

  • Concave Up

  • Concave Down.


Calculus-Derivative Example and Derivative Problems (Solved)

Let f(x) be a function where f(x) = x2

The derivative of x2 is 2x means that with every unit change in x, the value of the function becomes twice (2x).


Concept Of Limits and Derivatives to Solve Derivative Problems

When dx is made to be so little that is to become almost nothing. With Limits, we mean to say that x approaches zero but does not become zero.

Mathematically: means for all real ε > 0 there exists a real δ > 0 such that for all x with 0 < |x − c| < δ, we have |f(x) − L| < ε


Key Concepts to Solve Derivative Questions

  • To differentiate a power of x that is in the denominator, we have to first express it as a power with a negative exponent. Eg. \[\frac{1}{x^{2}}\] = x-2

  • The derivative rules make the process simple by differentiating polynomial functions.

  • To differentiate a radical, we have to first express it as a power with a rational exponent.


Apply Derivative Rules to Solve an Instantaneous Rate of Change Problems

Question) A skydiver jumps out of a plane from a height of 2200 m. The skydiver’s height above the ground, in meters, after t seconds is represented by the function h(t) = 2200 – 4.9t2 (assuming air resistance is not a factor). How fast is the skydiver falling after 4 s?

Solution) The instantaneous rate of change of the height of the skydiver at any point in a given time is shown by the derivative of the height function.

h(t) = 2200 – 4.9t2 

h'(t) = 0 – 4.9 (2t) = -9.8 t

Substitute t = 4 into the derivative function so as to find the instantaneous rate of change at a given time which is 4 s.

h'(t) = – 9.8 (4) = -39.2

After 4 s, the skydiver is falling at a rate of 39.2 m/s.

FAQ (Frequently Asked Questions)

Question 1) What is the Derivative Product Rule?

Answer) The product rule is defined if the two "parts" of the function are being multiplied together, and the chain rule is if they are being composed. For instance, to find the derivative of the function f(x) = x² sin(x), you use the product rule as they are being multiplied, and to find the derivative of g(x) = sin(x²) you use the chain rule.

Question 2) What is the Derivative Formula?

Answer) Derivatives are a fundamental tool of calculus. The derivative of a given function for a real variable measures the sensitivity to change of a given quantity, which is determined by another quantity. The derivative Formula is written as:-

 f1(x) = limx➝0f(x+Δx)-f(x) / Δx

Question 3) What is the power rule for Derivatives?

Answer) The power rule in calculus is a fairly simple rule that helps you find the derivative of a variable raised to a power, such as x5, 2x8, 3x-3, or 5x1/2. All we have to do is take the exponent and multiply it by the coefficient (the number in front of x), and decrease the exponent by 1.

Question 4) What is the Derivative of 0?

Answer) The derivative of 0 is 0. In general, we have the following rule for finding the derivative of a constant function, f(x) = a.

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