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Magazine Problem 113
Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sqrt{t})^{t}$$
Problem 114
Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=t^{\sqrt{t}}$$
Problem 115
Use the identity \(\csc ^{-1} u=\frac{\pi}{2}-\sec ^{-1} u\) to derive the formula for the derivative of \(\csc ^{-1} u\) in Table 7.3 from the formula for the derivative of \(\sec ^{-1} u\).
Problem 115
Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sin x)^{x}$$
Problem 116
Derive the formula \(\frac{d y}{d x}=\frac{1}{1+x^{2}}\) for the derivative of \(y=\tan ^{-1} x\) by differentiating both sides of the equivalent equation tan \(y=x\).
Problem 116
Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=x^{\sin x}$$
Problem 117
Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=\sin x^{x}$$
Problem 118
Use the identity \(\cot ^{-1} u=\frac{\pi}{2}-\tan ^{-1} u\) to derive the formula for the derivative of \(\cot ^{-1} u\) in Table 7.3 from the formula for the derivative of \(\tan ^{-1} u\).
Problem 118
Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\ln x)^{\ln x}$$
Problem 119
What is special about the functions \(f(x)=\sin ^{-1} \frac{x-1}{x+1}, \quad x \geq 0, \quad\) and \(\quad g(x)=2 \tan ^{-1} \sqrt{x} ?\) Explain.
