Mathematics, 02.08.2019 02:50 hjeffrey168

Find the derivative of the inverse to f(x)=x+3sin x at x=6

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Refresh if you see errors.disclaimer: i use approximations in my answer.not sure if x + 3sin(x) = 6 is solvable algebraically.======let $y=f(x)$ . then $y = x + 3\sin x$ the inverse relation of this can be obtained by swapping the x and y. you do not have to solve for y (that's impossible here, to my knowledge) $x = y + 3\sin y$ we can implicitly differentiate this equation with respect to x.this will give us the derivative for the inverse relation. $\begin{aligned} \dfrac{d}{dx}x & = \frac{d}{dx}(y + 3\sin y) \\ 1 & = \frac{d}{dx}(y) + \frac{d}{dx}(3\sin y) \\ 1 & = \frac{dy}{dx} + 3 \cos y \frac{dy}{dx} \\ 1 & = \frac{dy}{dx}(1 + 3\cos y) \\ \frac{dy}{dx} & = \frac{1}{1+3\cos y} \end{aligned}$ this derivative expression requires the y-coordinate at x=6 on the inverse relation.this next step requires some understanding the inverse.the domain of the inverse to f(x) is the range of f(x).the range of the inverse to f(x) is the domain of f(x).in other words, the domain and range swap places.therefore, we need to find the x-value of $f$ at $y=6$ to be able to get the y-value of the inverse function at x=6.to find the x-value of $f$ at $y=6$ , we solve this equation: $6=x + 3\sin x$ on a graphing calculator, graph $y=x + 3\sin x$ and $y=6$ .the intersection is approximately 6.21234, so that is the approximate solution.so on the graph of $f$ we have the point $(6.21234, 6)$ .therefore, on the graph of the inverse, we have the point $(6,6.21234)$ .this swap gets us the y-value of the inverse to $f$ at $x=6$ . so with $y=6.21234$ , we can calculate the derivative of the inverse t $x=6$ : $\begin{aligned} \frac{dy}{dx} & = \frac{1}{1+3\cos y} \\ & = \frac{1}{1+3\cos (6.21234)} \\ & \approx 0.25047 \end{aligned}$ therefore, the derivative of the inverse at x=6 is approximately 0.250.======i have attached an image, generated in desmos, of the inverse to f(x), using that approximate derivative for the tangent line.for this solution method, i assumed that the graphing calculator is unable to graph implicit relations like $x = y + 3\sin y$ (even though i use a graphing calculator that can do that to check). if you have access to graphing calculator that is able to graph out $x = y + 3\sin y$ , then just graph the inverse out, find the y-value of that inverse relation at x=6, and use that y-value for the derivative expression. that is another way of getting y=6.21234 at x=6 for the inverse of f(x).

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Find the derivative of the inverse to f(x)=x+3sin x at x=6...

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