I want to ask whether every point of a differentiable nonlinear function can have unique tangents with respect to its neighbouring points i.e every point of a differentiable function have different derivatives than its neighbouring points derivatives.
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1I guess $e^x$ is a basic example. Having different tangent at different point – parth sachdeva Jul 17 '22 at 13:18
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1What is the exact thing ,you want to ask – parth sachdeva Jul 17 '22 at 13:18
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$f(x)=x^4\sin^2(1/x),f(0)=0$ the tangent at $x=0$ is the same as the tangent at $x=1/(n\pi)$ – Empy2 Jul 17 '22 at 13:19
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@parthsachdeva Thanks ,you pointed out a perfect example to work with. – Anuj Jul 17 '22 at 13:20
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1@Anuj Actually even x^2 does have different tangents for different x – parth sachdeva Jul 17 '22 at 13:23
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Now that you mention it ,the function with form x^a does have different tangents for different value of x – Anuj Jul 17 '22 at 13:31