Differentiation Formulas

The derivative of a constant is zero.

> f1:=x->c;

> expr1:=(f1(x+h)-f1(x))/h;

Enough said! --- since .

Power Rule: if n is a positive integer, derivative of is .

> f2:=x->x^n;

> expr2:=(f2(x)-f2(a))/(x-a);

> limit(expr2,x=a);

> simplify(%);

Enough said. Just change the "a" to "x".

e.g.

> Diff(x^10,x)=diff(x^10,x);

Big Fact : The power rule applies to any n R we will prove this later in the course.

e.g.

> Diff(sqrt(x),x)=diff(sqrt(x),x);

Constant Multiple Rule: the derivative of is c f'( x ).

Proof: See the blackboard!

e.g.

> f2:=6*x^3;

> diff(f2,x);

> 6*diff(x^3,x);

Sum (difference) rule: the derivative of a sum (difference) is the sum (difference) of the derivatives. (f + g)' = f' + g' .

Proof: see blackboard.

e.g.

> f3:=6*x^3-x^10+7;

> diff(f3,x);diff(6*x^3,x)-diff(x^10,x)+diff(7,x);

Product Rule: (fg)' = f'g + fg'.

Proof: See blackboard.

> f4:=x^2*(x^3+5);

> diff(f4,x);

> simplify(%);

Quotient Rule: ( )' = .

> f5:=sqrt(x)/(x^2+1);

> diff(f5,x);

> simplify(%);

> ((Diff(sqrt(x),x)*(x^2+1))-Diff(x^2+1,x)*sqrt(x))/((x^2+1)^2);

> simplify(%);

> value(%);

See if you can do it!!

E.G.'s

1.

Find the points on the curve where the tangent is horizontal.

> eq1:=x^3-x^2-x+1;

> eq2:=diff(eq1,x);

> solve(eq2=0);

> subs(x=-1/3,eq1);subs(x=1,eq1);

> plot({eq1,32/27,0},x=-2..2,y=-4..4,colour=black);

2.

The normal line to y = f(x) at P is the line that passes through P and is perpendicular (normal) to the tangent line to y = f(x) at P. Find the normal line to at (-8,-2). Sketch

> eq3:=surd(x,3);

> slope:=diff(eq3,x);

> subs(x=-8,slope);

> simplify(%);

==> The slope of the normal line is 1/(1/12) = 12

> normal_line:=y-(-2)=-12*(x-(-8));tangent_line:=y-(-2)=(1/12)*(x-(-8));

> line1:=solve(normal_line,y);line2:=solve(tangent_line,y);

> plot({eq3,line1,line2},x=-14..0,y=-4..0,colour=black,numpoints=5000);

>

3.

Show that the curve y = has no tangent line with slope 4.

> curve:=6*x^3+5*x-3;

> slope:=diff(curve,x);

> solve(slope=4);

> plot({curve,slope},x=-2..2,y=-12..12,colour=black);