Chapter 3: Problem 2
How do you find the derivative of the quotient of two functions that are differentiable at a point?
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a. Use derivatives to show that \(\tan ^{-1} \frac{2}{n^{2}}\) and \(\tan ^{-1}(n+1)-\tan ^{-1}(n-1)\) differ by a constant. b. Prove that \(\tan ^{-1} \frac{2}{n^{2}}=\tan ^{-1}(n+1)-\tan ^{-1}(n-1)\)
Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x\), find a formula for \(\frac{d^{2}}{d x^{2}}(f(x) g(x))\)
Horizontal tangents The graph of \(y=\cos x \cdot \ln \cos ^{2} x\) has seven horizontal tangent lines on the interval \([0,2 \pi] .\) Find the approximate \(x\) -coordinates of all points at which these tangent lines occur.
Let \(f\) and \(g\) be differentiable functions with \(h(x)=f(g(x)) .\) For a given constant \(a,\) let \(u=g(a)\) and \(v=g(x),\) and define $$H(v)=\left\\{\begin{array}{ll} \frac{f(v)-f(u)}{v-u}-f^{\prime}(u) & \text { if } v \neq u \\ 0 & \text { if } v=u. \end{array}\right.$$ a. Show that \(\lim _{x \rightarrow u} H(v)=0\) b. For any value of \(u\) show that $$f(v)-f(u)=\left(H(v)+f^{\prime}(u)\right)(v-u).$$ c. Show that. $$h^{\prime}(a)=\lim _{x \rightarrow a}\left(\left(H(g(x))+f^{\prime}(g(a))\right) \cdot \frac{g(x)-g(a)}{x-a}\right).$$ d. Show that \(h^{\prime}(a)=f^{\prime}(g(a)) g^{\prime}(a)\).
Find \(\frac{d y}{d x},\) where \(\left(x^{2}+y^{2}\right)\left(x^{2}+y^{2}+x\right)=8 x y^{2}\)
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