Subsections

• Introduction
• D and diff
• Tangent Lines
• Exercises

Derivatives

Introduction

The purpose of this lab is to calculate derivatives using the diff command and the D command in Maple.

D and diff

Maple knows how to take many derivatives. Its main commands for doing this are D and diff. D is designed to differentiate functions, whereas diff is for differentiating expressions. However, if proper notation is used, diff can also be used with functions. To review the difference between a function and an expression, see the two examples below. The $f$ statement defines a function, the $g$ statement defines an expression.

> f:=x->x^2-3*x+1;
> g:=x^2+5*x-2;

Here are some examples that show how D and diff work for the function $f$ and the expression $g$.

> D(f);
> D(f)(x);
> diff(f(x),x);
> diff(g,x);

Note that the output for D(f) is the only one above that has an arrow in it. This means that if you assign the output of D(f) to a label, the derivative can be used as a function in Maple whereas all other outputs when given a label would have to be used as expressions. The examples below show how this works for $f'(1)$ and $g'(1)$.

> df := D(f);
> df(1);
> df2 := D(f)(x);
> subs(x=1, df2);
> df3 := diff(f(x),x);
> subs(x=1, df3);

The D command and the diff command do not require the function or expression to be entered first. Here are some other examples of how these commands can be used.

> D(sin);
> D(sin)(x);
> D(sin)(Pi/2);
> diff(cos(x),x);

Tangent Lines

Finding the equation of a tangent line to a function at a given point $x=x_0$ can be done using the point-slope form of a line.

$$y-y_0=m(x-x_0)$$

Where $y_0=f(x_0)$ and $m = f'(x_0)$.

$$y_{tanline} = f(x_0) + f'(x_0)(x-x_0)$$

This can be done rather easily in Maple and we can even plot $f(x)$ and the tangent line on the same graph. See how this can be done for the tangent line to $f(x)=\sqrt{x^2-x+1}$ at $x=1$.

> f := x-> sqrt(x^2-x+1);
> y_tan := f(1) + D(f)(1)*(x-1);
> plot({f(x),y_tan}, x=0..2);

Exercises

1. Compute the derivatives of the following functions using both the diff command and the D operator for each function.
   1. $$f(x)= \frac{\cos (x^2)}{tan(x)+1}$$
   2. $$g(x)= (x^6+3x^4-5)^3(-x^2-19x+5)^4$$
2. Find the equation of the tangent line to $$f(x) = \frac{\sqrt{x}}{3} \cos(x)$$ at the point where $x=6$. Include a plot of the function and the tangent line on the same graph.
3. Find all points on the graph of $h(x)= -x^3+7x-12$ where the slope is equal to -1. Don't forget that a point includes both the $x$ and $y$ coordinates.