csat-suneung 2025 Q27C

csat-suneung · South-Korea · csat__math 3 marks Composite & Inverse Functions Derivative of an Inverse Function

For a cubic function $f(x)$ with leading coefficient 1, let the function $g(x)$ be $$g(x) = f\left(e^{x}\right) + e^{x}$$ The tangent line to the curve $y = g(x)$ at the point $(0, g(0))$ is the $x$-axis, and the function $g(x)$ has an inverse function $h(x)$. What is the value of $h'(8)$? [3 points]
(1) $\frac{1}{36}$
(2) $\frac{1}{18}$
(3) $\frac{1}{12}$
(4) $\frac{1}{9}$
(5) $\frac{5}{36}$

For a cubic function $f(x)$ with leading coefficient 1, let the function $g(x)$ be
$$g(x) = f\left(e^{x}\right) + e^{x}$$
The tangent line to the curve $y = g(x)$ at the point $(0, g(0))$ is the $x$-axis, and the function $g(x)$ has an inverse function $h(x)$. What is the value of $h'(8)$? [3 points]\\
(1) $\frac{1}{36}$\\
(2) $\frac{1}{18}$\\
(3) $\frac{1}{12}$\\
(4) $\frac{1}{9}$\\
(5) $\frac{5}{36}$

Paper Questions

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