Thesecond derivative of a function is a derivative of the firstderivative of the function. To highlight this fact it is alsoreferred as second order derivative. As in case of firstderivatives, the second derivative notation also can be in differentforms. The prominent among them are Leibniz's notation and Lagrange'snotation.
Theformer is in the form (d2y/dx2),the number 2 in the notation not to be interpreted as a square. InLagrange’s form the derivatives are denoted by the functionfollowed by inverted commas, the number of commas indicate the orderof derivative. For example the second derivative of function f(x) isindicated as f’’(x).
Theconcept of second order derivative is very much used in analyzing thecritical points of the function. The first derivative of a functiongives us the critical points at which the value of the derivative is0. But it does not tell us whether it is a local maximum or localminimum or a point of inflection. But with the help of second orderderivative we can exactly figure out. Let us explain how it works.
Forany function f(x), the derivative f’(x) gives the slope of tangentat any point ‘x’. Suppose at x = c, the f’(c) = 0. It means theslope at x = c is 0, meaning the function is at local maximum valueor at a local minimum value.
Nowif it is a maximum then on the left to right the slope changes frompositive to negative. That means the rate of change of slope at x = cis negative which means that f”(c) is negative. Similarly, in caseof point of local minimum the slope changes from negative to positivein the direction left to right. So in this case f”(c) is positive.
Therefore,in general the second order derivative is negative at a point ofmaximum and positive at point of minimum. This concept is used totest the critical points for maxima and minima and hence this test iscalled as second derivative test.
Nowin some special cases, both f’(x) and f’’(x) also may be 0atthe same point. What does this indicate? Obviously the slope of thefunction at that point is 0 because f’(x) is 0 there. But sincef’’(x) is also 0 there, the slope does not change its sign.
Andhence the point is neither a point of maximum nor a point of minimum.Such a point is known as point of inflection where the concavity ofthe function changes, This analysis of concavity, due to its use, isdescribed as second derivative concavity.