91Ó°ÊÓ

Use logarithmic differentiation to find the derivative. $$f(x)=\left(x^{2}\right)^{4 x}$$

Prove that \(|x|<\left|\sin ^{-1} x\right|\) for \(0<|x|<1.\)

The functions tanh \(x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\) and \(\operatorname{coth} x=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}\) both have inverses on the appropriate domains and ranges. Show that \(\frac{d}{d x} \tanh ^{-1} x=\frac{1}{1-x^{2}}\) and \(\frac{d}{d x} \operatorname{coth}^{-1} x=\frac{1}{1-x^{2}} .\) Iden- tify the set of \(x\) -values for which each formula is valid. Is it fair to say that the derivatives are equal?

Find a general formula for the \(n\) th derivative \(f^{(n)}(x)\). $$f(x)=\sqrt{x}$$

Use the basic limits \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=1\) and \(\lim _{x \rightarrow 0} \frac{\cos x-1}{x}=0\) to find the following limits: (a) \(\lim _{x \rightarrow 0} \frac{\sin 3 x}{x}\) (b) \(\lim _{t \rightarrow 0} \frac{\sin t}{4 t}\) (c) \(\lim _{x \rightarrow 0} \frac{\cos x-1}{5 x}\) (d) \(\lim _{x \rightarrow 0} \frac{\sin x^{2}}{x^{2}}\)

Find a function \(g(x)\) such that \(g^{\prime}(x)=f(x)\). $$f(x)=\frac{x}{\sqrt{x^{2}+1}}$$

Suppose that \(f(t)\) represents the balance in dollars of a bank account \(t\) years after January \(1,2000 .\) Interpret each of the following. (a) \(\frac{f(4)-f(2)}{2}=21,034,\) (b) \(2[f(4)-f(3.5)]=\) 25,036 and \((\mathrm{c}) \lim _{h \rightarrow 0} \frac{f(4+h)-f(4)}{h}=30,000\)

If \(f^{\prime}(x)>0\) for all \(x,\) use the tangent line interpretation to argue that \(f\) is an increasing function; that is, if \(a

Use logarithmic differentiation to find the derivative. $$f(x)=x^{\ln x}$$

Use the product rule to show that if \(g(x)=[f(x)]^{2}\) and \(f(x)\) is differentiated, then \(g^{\prime}(x)=2 f(x) f^{\prime}(x) .\) This is an example of the chain rule, to be discussed in section 2.5