91Ó°ÊÓ

In Exercises \(1-57,\) find the derivatives. Assume that \(a, b,\) and \(c\) are constants. $$y=\frac{\sqrt{z}}{2^{z}}$$

Find the derivatives of the functions. Assume \(a, b,\) and \(c\) are constants. $$Q=\cos \left(e^{2 x}\right)$$

Find the derivative. It may be to your advantage to simplify before differentiating. Assume \(a, b, c\) and \(k\) are constants. $$f(z)=\frac{1}{\ln z}$$

Explain what is wrong with the statement. The function \(f(x)=\cosh x\) is periodic.

Find the derivative. It may be to your advantage to simplify before differentiating. Assume \(a, b, c\) and \(k\) are constants. $$g(t)=\frac{\ln (k t)+t}{\ln (k t)-t}$$

Find the derivatives of the functions. Assume \(a, b,\) and \(c\) are constants. $$h(t)=t \cos t+\tan t$$

Can the functions be differentiated using the rules developed so far? Differentiate if you can; otherwise, indicate why the rules discussed so far do not apply. $$y=e^{5 x}$$

Suppose \(f^{\prime}(x)\) is a differentiable decreasing function for all \(x\). In each of the following pairs, which number is the larger? Give a reason for your answer. (a) \(f^{\prime}(5)\) and \(f^{\prime}(6)\) (b) \(f^{\prime \prime}(5)\) and 0 (c) \(f(5+\Delta x)\) and \(f(5)+f^{\prime}(5) \Delta x\)

Find the derivatives of the given functions. Assume that \(a, b, c,\) and \(k\) are constants. $$y=\sqrt{x}(x+1)$$

In Exercises \(1-57,\) find the derivatives. Assume that \(a, b,\) and \(c\) are constants. $$y=\left(\frac{x^{2}+2}{3}\right)^{2}$$