Some Things to Remember from Cal I

J.Osborne.

The Definition of the Derivative

Recall: if is a function and x changes from x to x + h then the average rate of change of y with respect to x is given by . The (instantaneou s) rate of change of y with repsect to x is then given by making h closer and closer to zero by taking a limit:

The above is called the derivative of y with repsect to x and denoted as f'(x).

There is another slightly different way to get the derivative. If you let x change from x to a then the average rate of change of y is given by and we get the derivative by taking the limit as x appoaches a and obtain:

.

The above limit yields an expression in a (i.e. a function of a ) and to get f'(x) you just replace the a 's by x 's.

e.g. Find the derivative of using both definitions:

> f:=x->x^2;

First way

> quotient1:=(f(x+h)-f (x))/h;simplify(quotient1);

> Limit(quotient1,h=0):%=value (%);

===> f'(x) = 2x.

Second way

> quotient2:=(f(x)-f(a))/(x-a);simplify(quotient2);

> Limit(quotient2, x=a):%=value(%);

==> f'(a) = 2a so we get (changing a to x) that f'(x) = 2x as in the first way.

From the last eaxample we know that:

> fprime:=x->2*x;

> slope:=fprime(-2 );

> line:=y-4=slope*(x-(-2));

> Tangentline:=solve(line,y);

> plot({f(x),Tangentline },x=-4..2,thickness=2,colour=[red,blue]);

The Basic Rules of Differentiation

The Power rule:

The Constant Multiple rule: If k is a constant then

The Sum rule:

The Product r ule:

The Quotient rule:

The basic Trig Derivative formulas

The basic log and exponential derivatives:

e.g. Use the rules to verify the following:

A)

> f1:=7*x^4-sqrt(x)+8/x^4:Dif f(f1,x)=diff(f1,x);

B)

> f2:=x^2*sin(x):Diff(f2,x)=diff(f2,x);

C)

> f3:=(ex p(1)^x+sec(x))/x^(1/3):Diff(f3,x)=diff(f3,x);

1) If y = f(u) and u = g(x) (where f and g are differentiable functions) then y is a differentiable function of x and

.

2) If F(x) = f(g(x)) where f and g are differentiable functions then F'(x) = f'(g(x))g'(x)

General Power Rule : If f is any differentiable function and n N then = f'( x )

e.g. Verify the General Power Rule for

> y:=(x^3+1)^2;

Method 1: Expand and differentiate term-by-term :

> y_expanded:=expand(y);y_prime:=diff(y_expanded,x);

Method 2: Let Maple use the Chain Rule:

> y_prime:=diff(y,x);expand(%);

==> The answers are the same. You should check now by a hand calculation.

General Trig Derivatives :

Suppose that f is a differentiable function, then:

= cos(f(x)f'(x) = -sin(f(x))f'(x)

= f'(x) = sec(f(x))tan(f(x))f'(x)

= f'(x) = -csc(f(x))cot(f(x))f'(x)

e.g. Find the acceleration

Find the acceleration of a particle at t = 2 seconds if its position in meters from (0,0) at time t is given by . Realize: acceleration is the rate of change of velocity which, in turn, is the rate of change of position. Hence, acceleration is measured by the second derivative of position.

> s:=2*sin(2*t-3);

> velocity:=diff(s,t);

> evalf(subs(t=2,velocity));

> acceleration:=diff(velocity,t);

==> after 2 seconds the acceleration will be:

> subs(t=2,acceleration);

> evalf(%);

==> At 2 seconds, the particle will be moving at about 2 m/s and losing speed at the rate of about -6.7 m / .

?? Why are the acceleration units meters/ ??

General Logarithmic and Exponential Derivatives

If f is a differentiable function then:

and

and

e.g. Sketch the graph of

> eq:=exp(1)^(-x^2);eqprime:=diff(eq,x);eqdoubleprime:=diff(eqprime,x);

> solve(eqprime=0);solve(eqdoubleprime=0);

> plot(eq,x=-3..3,thickness=2);