If $g$ is the inverse of $f$ and $f'(x) = \frac{1}{1 + x^5}$,then $g'(x) =$

The inverse of the function $f(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} + 2$ is given by

Consider $f: \{1, 2, 3\} \rightarrow \{a, b, c\}$ and $g: \{a, b, c\} \rightarrow \{\text{apple, ball, cat}\}$ defined as $f(1)=a, f(2)=b, f(3)=c$ and $g(a)=\text{apple}, g(b)=\text{ball}, g(c)=\text{cat}$. Show that $f, g$ and $g \circ f$ are invertible. Find $f^{-1}, g^{-1}$ and $(g \circ f)^{-1}$ and show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

Let $f(x) = x^3 + 8x + 3$. Which one of the properties of the derivative enables you to conclude that $f(x)$ has an inverse?

The inverse of $y = 5^{\log x}$ is

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x) = 5x - 3$ and $g(x) = x^2 + 3$,then $g \circ f^{-1}(3)$ is equal to