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Chapter 3: Problem 5
Let \(f(x)=\sin x .\) What is the value of \(f^{\prime}(\pi) ?\)
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Tangency question It is easily verified that the graphs of \(y=x^{2}\) and
\(y=e^{x}\) have no points of intersection (for \(x>0\) ), and the graphs of
\(y=x^{3}\) and \(y=e^{x}\) have two points of intersection. It follows that for
some real number \(2
Calculating limits exactly Use the definition of the derivative to evaluate the following limits. $$\lim _{x \rightarrow e} \frac{\ln x-1}{x-e}$$
Find the following higher-order derivatives. $$\frac{d^{2}}{d x^{2}}\left(\log _{10} x\right)$$
\(F=f / g\) be the quotient of two functions that are differentiable at \(x\) a. Use the definition of \(F^{\prime}\) to show that $$ \frac{d}{d x}\left(\frac{f(x)}{g(x)}\right)=\lim _{h \rightarrow 0} \frac{f(x+h) g(x)-f(x) g(x+h)}{h g(x+h) g(x)} $$ b. Now add \(-f(x) g(x)+f(x) g(x)\) (which equals 0 ) to the numerator in the preceding limit to obtain $$\lim _{h \rightarrow 0} \frac{f(x+h) g(x)-f(x) g(x)+f(x) g(x)-f(x) g(x+h)}{h g(x+h) g(x)}$$ Use this limit to obtain the Quotient Rule. c. Explain why \(F^{\prime}=(f / g)^{\prime}\) exists, whenever \(g(x) \neq 0\)
General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{d x}\left(x^{10 x}\right)$$
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