91影视

In the text we noted that if $f\left(u\right(v\left(x\right)\left)\right)$ was a composition of three functions, then its derivative is $\frac{df}{dx}=\frac{df}{du}脳\frac{du}{dv}脳\frac{dv}{dx}$. Write this rule in 鈥減rime鈥� notation.

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible.

• the formal definition of the derivative of a function \(f\) at a point \(x = c\) (both \(z\rightarrow x\) form and \(h \rightarrow 0\) form)

Use limits to give mathematical definitions for:

(a) the slope of the line tangent to the graph of a function f at the point x = 4.

(b) the line tangent to the graph of a function f at the point x = 4.

(c) the instantaneous rate of change of a function f at the point x = 1.

(d) the acceleration at time t = 1.65 of an object that moves with position function s(t).

Suppose f is ant cubic polynomial function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$prove that coefficients of f a, b, c, d can be expressed in terms of values of f(x) and its derivatives at the point x=2

Let $l$be the line connecting two points $(a,f(a\left)\right)$and $(b,f(b\left)\right)$on the graph of a function $f$. What does this line $l$have to do with the average rate of change of $f$on the interval $[a,b],$and why?

If $A={\mathrm{蟺谤}}^2$,and A and r are both functions of time t,find$\frac{dA}{dt}$

Fill in the blank:$\left(kf\right)\text{'}\left(x\right)=\_\_\_\_\_\_\_\_.$

Find the derivatives of each of functions in Exercises 17鈥�44. In some cases it may be convenient to do some preliminary algebra.

$f\left(x\right)=\ln \left(x^2+1\right){\left(e^x\right)}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

Use the differentiation rules developed in this section to find

the derivatives of the functions

$f\left(x\right)=x^2+x(2-3x^2)$