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Subsections

# 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 statement defines a function, the 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 and the expression .
> 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 and .
> 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 can be done using the point-slope form of a line. Where and . This can be done rather easily in Maple and we can even plot and the tangent line on the same graph. See how this can be done for the tangent line to at .
> 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. 2. 2. Find the equation of the tangent line to at the point where . Include a plot of the function and the tangent line on the same graph.
3. Find all points on the graph of where the slope is equal to -1. Don't forget that a point includes both the and coordinates.   Next: About this document ... Up: lab_template Previous: lab_template
Dina Solitro
2001-11-27