If g is the inverse of f and f^{prime}(x)=fra | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given that $g(x)$ is the inverse of the function $f(x)$,we have $g(x) = f^{-1}(x)$.This implies $f(g(x)) = x$.Differentiating both sides with resp

Vedclass · Accepted Answer

$1+(g(x))^3$

Vedclass · Answer

(B) Given that $g(x)$ is the inverse of the function $f(x)$,we have $g(x) = f^{-1}(x)$.This implies $f(g(x)) = x$.Differentiating both sides with respect to $x$ using the chain rule,we get:$f^{\prime}(g(x)) \cdot g^{\prime}(x) = 1$Therefore,$g^{\prime}(x) = \frac{1}{f^{\prime}(g(x))}$ ... $(i)$Given $f^{\prime}(x) = \frac{1}{1+x^3}$,we substitute $x$ with $g(x)$ to get:$f^{\prime}(g(x)) = \frac{1}{1+(g(x))^3}$ ... $(ii)$Substituting $(ii)$ into $(i)$,we get:$g^{\prime}(x) = \frac{1}{\frac{1}{1+(g(x))^3}} = 1+(g(x))^3$.

If $g$ is the inverse of $f$ and $f^{\prime}(x)=\frac{1}{1+x^3}$,then $g^{\prime}(x)$ is

Similar Questions

Let $f: N \rightarrow R$ be a function defined as $f(x)=4x^{2}+12x+15$. Show that $f: N \rightarrow S$,where $S$ is the range of $f$,is invertible. Find the inverse of $f$.

If the function $f(x) = -4e^{\left(\frac{1-x}{2}\right)} + 1 + x + \frac{x^2}{2} + \frac{x^3}{3}$ and $g(x) = f^{-1}(x)$,then the value of $g'(-\frac{7}{6})$ equals

Let $f(x)=\sin 2x + \cos 2x$ and $g(x)=x^2-1$. Then $g(f(x))$ is invertible in the domain:

If $f(x) = \int\limits_0^x {\frac{1}{{\sqrt {1 + {t^3}} }}\,} dt$ and $h(x)$ is the inverse of $f(x)$,then the value of $\frac{{h''(x)}}{{{h^2}(x)}}$ is

Let $S = \{a, b, c\}$ and $T = \{1, 2, 3\}$. Find $F^{-1}$ of the following function $F$ from $S$ to $T$,if it exists: $F = \{(a, 3), (b, 2), (c, 1)\}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module