91影视

Suppose g, h, and j are differentiable functions with the values for the function and derivative given in the following table:

Use the table to calculate the values of the derivatives listed in Exercises 9鈥�16.

If $f\left(x\right)=h\left(g\left(x\right)j\left(x\right)\right)$, find$f\text{'}\left(0\right)$

In the text of this section we displayed graphs of f (x) = $x^4$and its first five derivatives. Use the slope-height behavior of the graphs to verify that each is the associated slope function of the one before.

For the graph of f shown next at the left, list the following quantities in order from least to greatest:

(a) the average rate of change of f on$[-1,1]$

(b) the instantaneous rate of change of f at$x=1$

(c) $f\text{'}(-1)$

(d) $\frac{f\left(2\right)-f(-1)}{2-(-1)}$

Express the sentence 鈥渢he derivative of $f\left(x\right)=3x^2-1$at $x=鈭�4$is $-24$鈥� in each of the following notations:

(a) prime (b) Leibniz (c) operator

Find the derivatives of the function:$f\left(x\right)=\frac{e^x}{\sin \left(x\right)}.$$f\left(x\right)=\frac{e^x}{\sin \left(x\right)}.$

Fill in the blank:

$\frac{d}{dx}\left(x^k\right)=---$ ($k$ is a constant)

The preceding figure at the right shows the graphs of y = tanh x, y = 1, and y = 鈭�1. For each of the statements that follows, explain graphically why the statement is true. Then justify the statement algebraically, using the definitions of the hyperbolic functions.

(a) 鈭�1 鈮� tanh x 鈮� 1

(b)$\underset{x鈫�鈭�}{\lim }\tan hx=1and\underset{x鈫�-鈭�}{\lim }\tan hx=-1.$

For the graph of g(x) shown next at the right, list the following quantities in order from least to greatest:
(a) the average rate of change of g on $[0,1]$
(b) the instantaneous rate of change of g at $x=1$is $0.4$
(c) $\frac{g(-1+0.1)-g(-1)}{0.1}$

(d) $\frac{g\left(1\right)-g(-1)}{1-(-1)}=$

Suppose g, h, and j are differentiable functions with the values for the function and derivative given in the following table:

Use the table to calculate the values of the derivatives listed in Exercises 9鈥�16.

If $f\left(x\right)=h\left(h\left(h\left(x\right)\right)\right)$, find$f\text{'}\left(1\right)$

Express the sentence 鈥渢he fourth derivative of $f\left(x\right)=3x^2-1$is equal to $0$鈥� in each of the following notations:

(a) prime (b) Leibniz (c) operator