91影视

For each of the equations in Exercises 59鈥�62, y is defined as an implicit function of x. Solve for y, and use what you find to sketch a graph of the equation.

$\frac{1}{4}{x}^2+y^2=9$

In Exercises 59鈥�63, find a function f that has the given derivative f'. In each case you can find the answer with an educated guess-and-check process.

$f\text{'}\left(x\right)=\frac{4e^{4x}(3x^5-1)-e^{4x}\left(15x^4\right)}{{\left(3x^5-1\right)}^2}$

Use the differentiation rules developed in this section to find the derivatives of the functions in Exercises 35-64. Note that it may be necessary to do some preliminary algebra before differentiating.

$f\left(x\right)=\frac{2x-3}{5x+4}$

Suppose $f$ is a linear function with a positive slope. Show that the average rate of change of $f$on an interval $[a,b]$ is positive, and then use this fact to show that $f$ is always increasing.

Show that $y\left(x\right)=4e^{3x}$is a solution of the differential

equation $\frac{dy}{dx}=3y$

Why does the proof of the power rule (Theorem 2.9) in this text work only when k is an integer? Also, at which point in the proof do we use the fact that k is positive?

Finding critical points: For each of the following functions $f$, find all of the $x$-values for which $f\text{'}\left(x\right)=0$and all of the $x$-values for which $f\text{'}\left(x\right)$does not exist.

$f\left(x\right)={\left(x\sqrt{x+1}\right)}^{-2}$.

Translate expressions written in Leibniz notation to 鈥減rime鈥� notation, and vice versa.

$\frac{d}{dx}\left(y\left(x\right)\right)$

Find the derivatives of$f\left(x\right)=\frac{1}{\sqrt{x}}.$

How is velocity different from speed? What does it mean if velocity is negative?