Calculating the Derivative of the Inverse Function Using the Chain Rule

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To find the derivative of the inverse function F^-1(x) at x = 1 for F(x) = e^(-x^2) + 3x, it is established that F(0) = 1, leading to F^-1(1) = 0. The derivative F'(x) is calculated as -2xe^(-x^2) + 3, but the initial calculation omitted the derivative of the constant term, "+3." The correct application of the chain rule shows that the derivative of the inverse function is given by 1/F'(F^-1(x)). The final result for the derivative at x = 1 is e/2, confirming the importance of including all terms in the derivative calculation.
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F(X) = e^-(x^2)+3x. Find D/DX (F^-1(x)) a x = 1.
Here's what i did, where is my mistake?
F(0)=1 therefore F^-1(1)=0
F'(x)= - 2xe^-(x^2)
d/dx (f^-1(x)) = 1/ F'(f^-1(x)) = 1/[-2(1)e^(-(1)^2)] = 1/2e^-1 = e / 2
 
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The derivative of F is incorrect.
 
-2x* e^(-X^2)
how is it incorrect?
 
The derivative of a sum is the sum of the derivatives. You haven't added the derivative of the second term in your function, F(x) = e-x2 + 3x.
 
You missed the "+3" part of it.

Daniel.
 
thank you very much
 

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