The problems of the acceleration of speed or velocity, marginal cost, rate of change etc are often asked in competitive examinations. If you will notice all these type of problems common factor is incremental change, i.e. a variable is established, so here we not discussing about constant speed or constant cost etc.
All such type of problems are solved using the concept of function. We can explain by an example of acceleration,
We know acceleration is rate of change of speed, while speed is usually used with concept of fixed time and distance, velocity is often used with concept of distance covered over a period time in a specific direction.
In case of velocity we denote the whole distance and time into small parts called delta often denoted by (d), thus
Acceleration = Rate of change of velocity at the small delta distance (dx) / Rate of change of time at the small delta time (dt)
Acc. = dx/ dt , This is called as derivative function where we can also represent Acc. = d [f(t)] /dt or f is a function of t.
In general terms often we use x and y and derivative function is denoted as y = f(x) where f(x) = d(x)/d(y)
This often denoted as y = f(x). Now there can be different way vairable can be set for e.g. y=x2 + x + 5
Rules for Derivatives
- If there is no change or the value of y remains constant then derivative function is denoted as zero. Obviously, if dy/dx denotes change and there is no change dy/dx = 0, initial equation being y = contant
- For the equations like y = xn , derivatives of such equations are represented as dy/dx = nxn-1
so for equations like y = x3 + 5x2 + 7x + 8 d(y)/ d(x) = 3 x2 + (5 x 2)x + 7 = 3x2 + 10x + 7
- if y = m x n ( where m and n are functions of x) then
- If y = m/ n then
- If y = kn where k is a function of x
- Derivatives of logs and exponential functions are used for as follows:
The above the rules will help you solve the problems that may be directly asked based on the above rules, there are some more practical applications of the derivatives used to solve the real life problems. These are as follows:
Velocity and Acceleration:
If x is distance travelled and t is time in seconds and the distance travelled is given by the equation of the form x = 7t2 + 6t + 8 then
Velocity = d(x)/dt and Acceleration = d2(x)/dt2
Marginal cost is the cost of each increments. If x is unit cost, then the total cost is T
Finding the tangent:
If y = f(x), the equation of the curve is y = x^3 + 8x^2 + 7, the d(y)/d(x) is the slope of the tangent to the curve.
Maximum and Minimum Values:
We can use the derivative equation to find the maximum and minimum values of the, this is clearly illustrated with the following example:
We will follow-up with some practice questions.
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