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Chapter 3: Problem 34
Use the General Power Rule where appropriate to find the derivative of the following functions. $$f(x)=x^{e}$$
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Derivatives and inverse functions $$\text { Find }\left(f^{-1}\right)^{\prime}(3) \text { if } f(x)=x^{3}+x+1$$
Find the derivative of the inverse of the following functions at the specified point on the graph of the inverse function. You do not need to find \(f^{-1}\) $$f(x)=x^{2}-2 x-3, \text { for } x \leq 1 ;(12,-3)$$
The Witch of Agnesi The graph of \(y=\frac{a^{3}}{x^{2}+a^{2}},\) where \(a\) is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi). a. Let \(a=3\) and find an equation of the line tangent to \(y=\frac{27}{x^{2}+9}\) at \(x=2\) b. Plot the function and the tangent line found in part (a).
Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of \(f^{-1}\) . $$f(x)=(x+2)^{2} ;(36,4)$$
The following limits equal the derivative of a function \(f\) at a point a. a. Find one possible \(f\) and \(a\) b. Evaluate the limit. $$\lim _{h \rightarrow 0} \frac{\cos \left(\frac{\pi}{6}+h\right)-\frac{\sqrt{3}}{2}}{h}$$
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