91Ó°ÊÓ

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \left(\frac{\sqrt{\sin \theta \cos \theta}}{1+2 \ln \theta}\right)$$

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=x^{2}+2 x, \quad x_{0}=1, \quad d x=0.1$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\tan ^{-1} \sqrt{x^{2}-1}+\csc ^{-1} x, \quad x>1$$

Find the derivatives of the functions in Exercises \(23-50\). $$h(x)=x \tan (2 \sqrt{x})+7$$

Do the graphs of the functions have any horizontal tangents in the interval \(0 \leq x \leq 2 \pi ?\) If so, where? If not, why not? Visualize your findings by graphing the functions with a grapher. $$y=x+\sin x$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \left(\frac{\left(x^{2}+1\right)^{5}}{\sqrt{1-x}}\right)$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$x^{2} \cos ^{2} y-\sin y=0, \quad(0, \pi)$$

Each function \(f(x)\) changes value when \(x\) changes from \(x_{0}\) to \(x_{0}+d x .\) Find a. the change \(\Delta f=f\left(x_{0}+d x\right)-f\left(x_{0}\right)\) b. the value of the estimate \(d f=f^{\prime}\left(x_{0}\right) d x ;\) and c. the approximation error \(|\Delta f-d f|\) (GRAPH CANNOT COPY) $$f(x)=2 x^{2}+4 x-3, x_{0}=-1, \quad d x=0.1$$

Find the derivatives of the functions in Exercises \(23-50\). $$k(x)=x^{2} \sec \left(\frac{1}{x}\right)$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cot ^{-1} \frac{1}{x}-\tan ^{-1} x$$