Question

How do you find \[\dfrac{{{d^2}y}}{{d{x^2}}}\] ?

Answer 1

Hint: When we differentiate an already differentiated function, the process is called double differentiation and the result obtained is called the second derivative of the given function. In simple words, the second derivative is used to find out how quickly the change of a quantity is changing. There are various uses of the second derivative; it tells us the concavity of a curve. The second derivative of a quantity x with respect to some other quantity y is represented as $ \dfrac{{{d^2}y}}{{d{x^2}}} $

Complete step-by-step answer:
Let a quantity be y that is changing with respect to x, then its instantaneous change/first derivative is given as $ \dfrac{{dy}}{{dx}} $ .
To find the second derivative, we differentiate the first derivative again with respect to x as shown – $ \dfrac{d}{{dx}}(\dfrac{{dy}}{{dx}}) $ , in simplified form this expression is written as $ \dfrac{{{d^2}y}}{{d{x^2}}} $ .
 For example – the instantaneous speed of a body is given as $ \dfrac{{dx}}{{dt}} $ , if we have to find the rate of change of speed, we will differentiate this expression again and get – $ \dfrac{{{d^2}x}}{{d{t^2}}} $
Hence, this way we can find $ \dfrac{{{d^2}y}}{{d{x^2}}} $ .

Note: Usually, we can observe a change in the given quantity over a specific duration of time, but for some quantities, we have to find the change in that quantity within a very short duration, that is we have to find the instantaneous change in the quantity. For finding the instantaneous change, we use differentiation, differentiation of a quantity y with respect to a quantity x is given as $ \dfrac{{dy}}{{dx}} $ where dy represents a very small change in y and x represents a very small change in x.