perplexus.info :: Calculus : An Inverse Evaluation

An Inverse Evaluation (Posted on 2006-05-25)

We know that f'(x) and f"(x) respectively denote the first derivative and second derivative of a given function f(x) with respect to x.
If P is the inverse function of S, and P'(x)=e^x²; then:

2*S(x)*S'(x) + e^S(x)²*S"(x) = ?

Submitted by K Sengupta

Rating: 4.0000 (2 votes)

Solution: (Hide)

The required missing quantity is 0.

EXPLANATION::(On basis of the solution submitted by JayDeeKay)

By definition, for each real x: P(S(x)) = x

Differentiating w.r.t x, using the Chain rule, we obtain:

P'(S(x))*S'(x) = 1

Accordingly, exp (S(x)^2) * S’(x) = 1 ..........(1)

Differentiating both sides of (1), using the Chain and Product rules, we obtain:

S”(x)* exp(S(x)^2) + S’(x)* exp (S(x)^ 2)*2*S(x) * S’(x) = 0

Consequently, from Eq (1), upon substitution, we obtain:

2*S(x)*S’(x) + exp (S(x)^2)* S”(x) = 0

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: Acknowledgement	JayDeeKay	2006-07-03 16:03:31
Acknowledgement	K Sengupta	2006-06-12 23:27:19
re(2): Maybe the way? yes it is!	Richard	2006-05-30 16:31:06
re: Maybe the way? yes it is!	JayDeeKay	2006-05-30 16:18:56
Maybe the way?	e.g.	2006-05-28 14:45:07
Starting idea	Gamer	2006-05-25 14:17:25

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