Limits, Continuity & Differentiability
Derivatives of Inverse Functions
Grade 12
Question:
<p><span style='color:red'>●Ex. 7</span> Let \(f(x) = e^{\log x}\). If \(g(x)\) is the inverse of \(f(x)\), then find \(g'(x)\).</p>
<p>(a) \(e^{e^x}\)</p>
<p>(b) \(e^x\)</p>
<p>(c) \(e^{e^x + x}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For inverse functions, if $y = f(x)$, find $x$ in terms of $y$ to get $g(y)$, then differentiate.
Step 1: Determine the function $f(x)$.
Given $f(x) = e^{\log x}$.
For $f(x)$ to be defined, we must have $x > 0$.
Using the property $e^{\log A} = A$, we simplify $f(x)$:
$f(x) = x$ for $x > 0$.
Step 2: Find the inverse function $g(x)$.
Let $y = f(x)$. Then $y = x$.
To find the inverse, we swap $x$ and $y$: $x = y$.
Thus, $g(x) = x$.
Step 3: Find the derivative of the inverse function $g'(x)$.
Since $g(x) = x$, its derivative is:
$g'(x) = \frac{d}{dx}(x) = 1$.
Correct Answer: C