91Ó°ÊÓ

Calculate the derivative of the following functions. $$y=\frac{1}{\log _{4} x}$$

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=\frac{x}{x+5}$$

Calculate the derivative of the following functions. $$y=\log _{2}\left(\log _{2} x\right)$$

The following limits represent \(f^{\prime}(a)\) for some function \(f\) and some real number a. a. Find a function \(f\) and a number \(a\). b. Find \(f^{\prime}(a)\) by evaluating the limit.. \(\lim _{h \rightarrow 0} \frac{(1+h)^{8}+(1+h)^{3}-2}{h}\)

Compute the derivative of the following functions. \(h(x)=\frac{(x+1)}{x^{2} e^{x}}\)

a. For the following functions, find \(f^{\prime}\) using the definition. b. Determine an equation of the line tangent to the graph of \(f\) at ( \(a, f(a)\) ) for the given value of \(a\) $$f(x)=\frac{1}{x} ; a=-5$$

Use the Chain Rule combined with other differentiation rules to find the derivative of the following functions. $$y=\left(\frac{3 x}{4 x+2}\right)^{5}$$

Find \(d y / d x\) for the following functions .$$y=\frac{x \cos x}{1+x^{3}}$$

Identifying functions from an equation The following equations implicitly define one or more functions. a. Find \(\frac{d y}{d x}\) using implicit differentiation. b. Solve the given equation for \(y\) to identify the implicitly defined functions \(y=f_{1}(x), y=f_{2}(x), \ldots\) c. Use the functions found in part (b) to graph the given equation. $$y^{2}=\frac{x^{2}(4-x)}{4+x} \text { (right strophoid) }$$

The following limits represent \(f^{\prime}(a)\) for some function \(f\) and some real number a. a. Find a function \(f\) and a number \(a\). b. Find \(f^{\prime}(a)\) by evaluating the limit.. \(\lim _{x \rightarrow 1} \frac{x^{100}-1}{x-1}\)