COMEDK · Maths · 26. Differentiation
If \(y=x+e^x\) then \(\dfrac{d^2 x}{d y^2}=\)
- A \(\dfrac{-1}{\left(1+e^x\right)^3}\)
- B \(e^x\)
- C \(\dfrac{-e^x}{\left(1+e^x\right)^2}\)
- D \(\dfrac{-e^x}{\left(1+e^x\right)^3}\)
Answer & Solution
Correct Answer
(D) \(\dfrac{-e^x}{\left(1+e^x\right)^3}\)
Step-by-step Solution
Detailed explanation
Given \(y = x + e^x\). Differentiating both sides with respect to \(x\), we get \(\dfrac{dy}{dx} = 1 + e^x\). Therefore, \(\dfrac{dx}{dy} = \dfrac{1}{\dfrac{dy}{dx}} = \dfrac{1}{1 + e^x} = (1 + e^x)^{-1}\). Now, differentiating \(\dfrac{dx}{dy}\) with respect to \(y\) using the…
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