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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Differentiation

NDA 2021 · Mathematics · Differentiation

What is the derivative of e^x with respect to x^e ?

What is the derivative of exe^x with respect to xex^e ?

Options

  1. Axexexe\frac{x e^x}{e x^e}
  2. Bexxe\frac{e^x}{x^e}
  3. Cxexxe\frac{x e^x}{x^e}
  4. Dexexe\frac{e^x}{e x^e}

Correct answer

A. xexexe\frac{x e^x}{e x^e}

Step-by-step solution

Let y=ex,dydx=exy=e^x, \frac{d y}{d x}=e^x and z=xe⇒dzdx=e⋅x(e−1)Now, dydz=dy/dxdz/dx=exe⋅x(e−1)=x⋅exe⋅xe\begin{aligned} & \text{ and } z=x^e \Rightarrow \frac{d z}{d x}=e \cdot x^{(e-1)} \\ & \text{Now, } \frac{d y}{d z}=\frac{d y / d x}{d z / d x}=\frac{e^x}{e \cdot x^{(e-1)}}=\frac{x \cdot e^x}{e \cdot x^e}\end{aligned}$$
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