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Derivatives are used widely in the fields of physics, maths, and chemistry. They have many uses and can solve many important questions. To find a derivative of anything means to find its rate of change.
What are Derivatives?
The rate of change of any vector value is called it’s derivative. If you want to see derivatives of any value, you can observe the change in the slope of any graph.
For example, the rate of change of the position of an object is called its velocity, and the rate of change of velocity is called acceleration. So, the derivative of a position gives velocity, and the derivative of velocity gives acceleration. Thus, you can find the first, second, third derivative, and so on. There are many complex formulae of derivatives. The rate of change of any object means how slow or fast it changes its position. The slope of the graph represents this transformation.
Let y be the equation of line, y = f(x) = mx + b
Here m is the slope.
The slope m will be calculated by m = change in y/ change in x.
The example of the derivative of a simple function is: f(x) = x2 is f’(x)= 2x and f’’(x)= 2.
The derivative of a quantity or a value gives you the change at an instant, at a specific place, and for a specific period of time.
Applications of Derivatives
Derivatives are mostly used in kinematics. Kinematics is a field of physics that deals with the study of motion. It is also present in calculus.
In kinematics, derivatives show how the motion of an object is related to others.
Velocity is the speed of a motion or an object in a given direction. Velocity can be written as a derivative of the displacement vector.
Let the displacement vector of an object be x (t) = 6x2 at time t. Then the velocity of that object can be written as the derivative of the displacement vector. x’(t)= 12x. This is the velocity of the object.
The rate of change of velocity is called acceleration at a specific time. It can also be said that the increase or decrease in the speed of the object is known as acceleration.
Let the velocity be v (t) = 2x-4x3 at any time t. Then the acceleration will be its derivative. v’(t)= 2-12x2.
Displacement, velocity, and acceleration are related to each other by their derivatives. It can also be written as: a (t) =v’ (t) =x’’ (t).
This equation can also be written as: a (t) = dv/dt = d2x/dt2.
Derivatives of Some Common Functions
Common functions | function | Derivative |
constant | c | 0 |
Line | x | 1 |
square | X2 | 2x |
Multiply by constant | cf | cf’ |
Power rule | xn | nxn-1 |
Sum rule | f + g | f’ + g’ |
Difference rule | f – g | f’ – g’ |
Product rule | fg | f’g + g’f |
Quotient rule | f/g | (f’g – g’f)/ g2 |
Reciprocal rule | 1/ f | -1/ f2 |
line | ax | a |
trigonometry | Sin(x) | cos(x) |
cos(x) | -sin(x) | |
tan(x) | sec2(x) | |
logarithms | ln (x) | 1/x |
| Loga(x) | 1/(x ln(a)) |
chain rule | F(g(x)) | F’(g(x))g’(x) |
Important Questions on Derivatives
- Let a particle in space move with a velocity v(t) = t2-2t, at any time t in seconds and velocity in m/s.
- a) When will the particle rest?
- b) Calculate the acceleration of a particle at t = 4 seconds.
- When a football is kicked in the air the path of the flight of the football can be given by a function x (t) = -9.8t2 + 4.9t + 5
- a) Calculate the acceleration at t = 5 s.
- b) Calculate the maximum height of the football and the time of its occurrence.
- c) What will be the time when the ball lands back on the ground?
- If a car’s displacement is represented by the function p(x) = 4x3-2x2,
- a) What will be the car’s velocity at x = 5 m/s?
- b) What will be the car’s acceleration at x = 1 m/x?
- The function represents the growth of the population of a virus p (t) = 6,000 + 3000t2. Here ‘t’ represents time in hours. Determine the following:
- a) The average rate of growth
- b) The growth at an instantaneous rate
- c) The growth at instant t = 5 hours.
- Some gas is put inside a cylindrical tank at 7 m3/ min. Let the pressure be constant; what will be the velocity when the radius changes at a diameter of 40 cm?
- State the differential coefficient of ax + log(x). Cos(x).
Conclusion
Isaac Newton and Gottfried Wilhelm Leibniz discovered the concept of Derivatives between the 1600s and 1700s. They individually came up with the concepts of derivations. Derivatives are widely used in calculus. They are used to find vector quantities like velocity, acceleration, momentum, etc. Derivatives are used to derive many formulae and are the basis of many derivations. There are many different rules and formulas to learn before you do differential function derivatives.
The basic representation of the relation between the derivatives of the position vector, velocity, and acceleration can be seen by a (t) =v’ (t) =x’’ (t). Derivatives are used to find the rate of change of any quantity over any particular period or instant in time. Also, in maths, the slope of the graph is a representation of derivatives.
