Rules for finding derivatives.
In this worksheet we will use the power/sum/difference/product/quotient
rules to find derivative of some functions. The results will then be checked
against Maple command diff.
First, we recall the followings (D stands for Dx , the derivative operator)
D(xn ) = n x(n−1)
D(k f(x)) = k Df(x) if k is a constant
D(f(x) + g(x)) = Df(x) + Dg(x)
D(f(x) − g(x)) = Df(x) − Dg(x)
D(f(x) g(x)) = Df(x) g(x) + f(x) Dg(x)
f(x)
D( g(x)
)=
Df(x) g(x)−f(x) Dg(x)
g(x)2
Rules for Trigonometric functions
D(sin(x)) = cos(x)
D(cos(x)) = −sin(x)
D(tan(x)) =
1
cos(x)2
= 1 + tan(x)2
1
2
D(cotan(x)) = − sin(x)
2 = −1 − cotan(x)
1
Sum rule
Compute the derivative of
>
f:=x->x^3+2/x;
f := x → x3 + 2
1
x
Using the power rule, we have
>
diff(x^3,x);
3 x2
Writing = x(−1) and using the power rule, we have
>
diff(x^(-1),x);
1
− 2
x
Putting together, we have the derivative of f(x) is 3 x2 + 2 (− x12 ). Check
with Maple, we have
>
diff(f(x),x);
1
3 x2 − 2 2
x
1
x
1
2
Product rule
Compute the derivative of
>
f:=x->(3*x^2-2*x)*(x^3+x^2);
f := x → (3 x2 − 2 x) (x3 + x2 )
Think of f(x) as the product of the following functions
>
p:=x->3*x^2-2*x; q:=x->x^3+x^2;
p := x → 3 x2 − 2 x
q := x → x3 + x2
We can use the power and sum rules to compute their derivatives and get
>
deriv_of_p:=diff(p(x),x); deriv_of_q:=diff(q(x),x);
deriv of p := 6 x − 2
deriv of q := 3 x2 + 2 x
>
deriv_of_f:=deriv_of_p * q(x) + p(x)*deriv_of_q;
deriv of f := (6 x − 2) (x3 + x2 ) + (3 x2 − 2 x) (3 x2 + 2 x)
Using Maple to check our result
>
diff(f(x),x);
(6 x − 2) (x3 + x2 ) + (3 x2 − 2 x) (3 x2 + 2 x)
We can see that they are identical. Let’s ask Maple to simplify the above
>
simplify(%);
15 x4 + 4 x3 − 6 x2
3
Quotient rule
Compute the derivative of
>
f:=x->(3*x-2)/(x^2+1);
3x − 2
x2 + 1
Think of f(x) as the quotient of the following functions
>
p:=x->3*x-2; q:=x->x^2+1;
p := x → 3 x − 2
q := x → x2 + 1
We can use the power and sum rules to compute their derivatives and get
>
deriv_of_p:=diff(p(x),x); deriv_of_q:=diff(q(x),x);
deriv of p := 3
deriv of q := 2 x
>
deriv_of_f:=(deriv_of_p * q(x) - p(x)*deriv_of_q)/(q(x))^2;
3 x2 + 3 − 2 (3 x − 2) x
deriv of f :=
(x2 + 1)2
Using Maple to check our result
>
diff(f(x),x);
(3 x − 2) x
1
−2
3 2
x +1
(x2 + 1)2
f := x →
2
Doesn’t look like ours! Let’s ask Maple to simplify the above
>
simplify(%);
3 x2 − 3 − 4 x
−
(x2 + 1)2
and simplify what we got
>
simplify(deriv_of_f);
3 x2 − 3 − 4 x
−
(x2 + 1)2
We can see that the two formulas are identical.
4
Derivative of Trig Functions
In this section we will apply the rules for finding derivative and the derivative
formulas for elementary trigonometric functions.
4.1
example 1
f:= x->(x^2-2*x)*(sin(x)+2*cos(x));
f := x → (x2 − 2 x) (sin(x) + 2 cos(x))
Think of f as the product of two functions
>
p:=x->x^2-2*x; q:=x->sin(x)+2*cos(x);
p := x → x2 − 2 x
q := x → sin(x) + 2 cos(x)
Derivative of p, as a polynomial, is easily computed by using the power rule.
>
deriv_of_p:=diff(p(x),x);
deriv of p := 2 x − 2
Derivative of q, as the sum of two Trig. functions, is easily computed by
using the sum rule and the Trig formulas.
>
deriv_of_q:=diff(q(x),x);
deriv of q := cos(x) − 2 sin(x)
Finally, by the product rule, we have the derivative of f(x) as
>
deriv_of_f := diff(p(x),x)*q(x)+p(x)*diff(q(x),x);
deriv of f := (2 x − 2) (sin(x) + 2 cos(x)) + (x2 − 2 x) (cos(x) − 2 sin(x))
Using Maple, we also have
>
diff(f(x),x);
(2 x − 2) (sin(x) + 2 cos(x)) + (x2 − 2 x) (cos(x) − 2 sin(x))
The results are the same.
>
4.2
example 2
f:= x->(x^2-2*x)*(sin(x)+2*cos(x));
f := x → (x2 − 2 x) (sin(x) + 2 cos(x))
Think of f as the product of two functions
>
p:=x->x^2-2*x; q:=x->sin(x)+2*cos(x);
p := x → x2 − 2 x
>
3
q := x → sin(x) + 2 cos(x)
Derivative of p, as a polynomial, is easily computed by using the power rule.
>
deriv_of_p:=diff(p(x),x);
deriv of p := 2 x − 2
Derivative of q, as the sum of two Trig. functions, is easily computed by
using the sum rule and the Trig formulas.
>
deriv_of_q:=diff(q(x),x);
deriv of q := cos(x) − 2 sin(x)
Finally, by the product rule, we have the derivative of f(x) as
>
deriv_of_f := diff(p(x),x)*q(x)+p(x)*diff(q(x),x);
deriv of f := (2 x − 2) (sin(x) + 2 cos(x)) + (x2 − 2 x) (cos(x) − 2 sin(x))
Using Maple, we also have
>
diff(f(x),x);
(2 x − 2) (sin(x) + 2 cos(x)) + (x2 − 2 x) (cos(x) − 2 sin(x))
The results are the same.
>
f:= x->(x^2-3*cos(x))/(sin(x));
x2 − 3 cos(x)
f := x →
sin(x)
Think of f as the product of two functions
>
p:=x->x^2-3*cos(x); q:=x->sin(x);
p := x → x2 − 3 cos(x)
q := sin
Derivative of p, as a sum, is easily computed by using the power rule and
Trig rule.
>
deriv_of_p:=diff(p(x),x);
deriv of p := 2 x + 3 sin(x)
Derivative of q is easily computed by using the Trig formulas.
>
deriv_of_q:=diff(q(x),x);
deriv of q := cos(x)
Finally, by the quotient rule, we have the derivative of f(x) as
>
deriv_of_f := (diff(p(x),x)*q(x)-p(x)*diff(q(x),x))/(q(x))^2;
(2 x + 3 sin(x)) sin(x) − (x2 − 3 cos(x)) cos(x)
deriv of f :=
sin(x)2
Using Maple, we also have
>
diff(f(x),x);
2 x + 3 sin(x) (x2 − 3 cos(x)) cos(x)
−
sin(x)
sin(x)2
The results are the same (Check it).
4
4.3
example 3
f:= x->(cos(x)-2*tan(x))/(cot(x)+1);
cos(x) − 2 tan(x)
f := x →
cot(x) + 1
Think of f as the quotient of two functions
>
p:=x->cos(x)-2*tan(x); q:=x->cot(x)+1;
p := x → cos(x) − 2 tan(x)
q := x → cot(x) + 1
Derivative of p, as a sum, is easily computed by using the Trig formulas.
>
deriv_of_p:=diff(p(x),x);
deriv of p := −sin(x) − 2 − 2 tan(x)2
Derivative of q, as the sum of a Trig. functions and a constant, is easily
computed by using the sum rule and the Trig formulas.
>
deriv_of_q:=diff(q(x),x);
deriv of q := −1 − cot(x)2
Finally, by the quotient rule, we have the derivative of f(x) as
>
deriv_of_f := (diff(p(x),x)*q(x)-p(x)*diff(q(x),x))/(q(x))^2;
(−sin(x) − 2 − 2 tan(x)2 ) (cot(x) + 1) − (cos(x) − 2 tan(x)) (−1 − cot(x)2 )
deriv of f :=
(cot(x) + 1)2
Using Maple, we also have
>
diff(f(x),x);
−sin(x) − 2 − 2 tan(x)2
(cos(x) − 2 tan(x)) (−1 − cot(x)2 )
−
cot(x) + 1
(cot(x) + 1)2
The results are actually the same.
>
5