Tutorial on Derivatives
A. derivative of a function of one variable.
The basic form of the derivitave command is D. It is used as
D[function,{var,n}] where n is n^th derivative of the function with respect
to a single variable.
Example1.
Define f(x)= x^2. Take the 1st, 2nd and the third derivative.
Solution:
Please type the following into Mathematica notebook and hit the enter
key.
f[x_]=x^2
d1=D[f[x],{x,1}] (*this gives the first derivative*)
d2=D[f[x],{x,2}] (*2nd derivative*)
d3=D[f[x],{x,3}] (*3rd derivative*)
(*Remember d1,d2 and d3 are names that you choose.*)
2
x
2 x
2
0
Example 2.
Define g(x)=cos kx where k is a constant. Take the 1st derivative .
Solution :
Type in
g[x_]=Cos[k x]
d4=D[g[x],{x,1}]
Cos[k x]
-(k Sin[k x])
Example 3.
Differentiate f(x)=cot(sqrt(x^2+1)).
Solution :
Type in
f[x_]=Cot[Sqrt[x^2+1]]
d5=D[f[x],{x,1}]
2
Cot[Sqrt[1 + x ]]
2 2
x Csc[Sqrt[1 + x ]]
-(--------------------)
2
Sqrt[1 + x ]
B. Derivative of the function of several variables.
The derivative f(x,y) with respect to x and y is D[f(x,y], {var,n}]
where n is nth partial derivative.
Example4
Find the derivative of
f(x,y)=x^3 +y^3 once with respect to x and then once with respect to y
Solution:
Type in
f[x,y]=x^3+y^3
d1=D[f[x,y],{x,1}] (*once with respect to x*)
d2=D[f[x,y],{y,1}] (*once with respect to y*)
3 3
x + y
2
3 x
2
3 y
Example 5.
Take the 2nd partial derivative of f(x,y) with respect to x
Solution
Type in
d3=D[f[x,y],{x,2}] (*twice with respect to x*)
6 x
Example 6
Find the third derivative of f(x,y) with respect to y
Solution
Type in
Clear[f,x,y] (* This clears all of the previous variable*)
f[x_,y_]=x^3+y^3 Cos[k Sqrt[y]]
d4=D[f[x,y],{y,3}]
3 3
x + y Cos[k Sqrt[y]]
2
15 k y Cos[k Sqrt[y]]
6 Cos[k Sqrt[y]] - ---------------------- -
8
3 3/2
57 k Sqrt[y] Sin[k Sqrt[y]] k y Sin[k Sqrt[y]]
--------------------------- + ----------------------
8 8