Let $f(x)$ be a function with continuous derivative such that $f(0)=4$ and $f’(0)=5$. Define $g(x)=f(x)+\sin {x}$, and let $h$ be the inverse of $g$. Find $h’(4)$.

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BlackTom AI · Accepted Answer

We start by identifying the relationship between g and f. Since g(x) = f(x) + sin x, we compute g(0) = f(0) + sin(0) = 4 + 0 = 4, so the input 4 corresponds to x = 0 under g, meaning h(4) = 0 because h is the inverse of g.

Next, use the formula for the derivative of an inverse function: if h is the inverse of g, then h'(y) = 1 / g'(h(y)). Evaluating at y = 4 gives h'(4) = 1 / g'(h(4)) = 1 / g'(0).

We compute g'(x) by differentiating g(x) = f(x) + sin x: g'(x) = f'(x) + cos x. At x = 0, f'(0) = 5 and cos(0) = 1, so g'(0) = 5 + 1 = 6.

Therefore, h'(4) = 1 / 6.

If you like a quick summary: locate the preimage of 4 under g (which is x = 0), differentiate g to get g'(0) = 6, and apply the inverse derivative formula to obtain h'(4) = 1/6.

Let \(f(x)\) be a function with continuous derivative such that \(f(0)=4\) and \(f’(0)=5\). Define \(g(x)=f(x)+\sin {x}\), and let \(h\) be the inverse of \(g\). Find \(h’(4)\).简答题

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