Implicit Differentiation made easy with Mathematica

PeterCullenBurbery
I work through various examples of implicit differentiation.

Details

These examples are from the documentation. I modified some of the functions and created my own examples.
In[]:=
q1=QuestionObject["What is the derivative of 2z with respect to x where z is constrained by
2
x
+
2
z
=1?",AssessmentFunction[ImplicitD[2z,
2
x
+
2
z
==1,z,x]]]
Out[]=
What is the derivative of 2z with respect to x where z is constrained by
2
x
+
2
z
=1?
​
​
Submit
In[]:=
ImplicitD[2z,
2
x
+
2
z
==1,z,x]
Out[]=
-
2x
z
In[]:=
q1
Out[]=
What is the derivative of 2z with respect to x where z is constrained by
2
x
+
2
z
=1?
-
2x
z
​
Submit
In[]:=
q2=QuestionObject["What is the second derivative of log(xz)​, assuming that
2
z
-xsin(z)0?",AssessmentFunction[ImplicitD[Log[xz],
2
z
-xSin[z]==1,z,{x,2}]]]
Out[]=
What is the second derivative of log(xz)​, assuming that
2
z
-xsin(z)0?
​
​
Submit
In[]:=
ImplicitD[Log[xz],
2
z
-xSin[z]==1,z,{x,2}]//TraditionalForm
Out[]//TraditionalForm=
1
2
x
2
z
3
(xcos(z)-2z)
(-
3
x
2
z
3
cos
(z)+
3
x
z
3
sin
(z)+2
3
x
zsin(z)
2
cos
(z)-
3
x
2
sin
(z)cos(z)+6
2
x
3
z
2
cos
(z)-4
2
x
2
z
sin(z)cos(z)+4
2
x
z
2
sin
(z)-12x
4
z
cos(z)+8
5
z
)
In[]:=
q2
Out[]=
What is the second derivative of log(xz)​, assuming that
2
z
-xsin(z)0?
8
5
z
-12x
4
z
Cos[z]+6
2
x
3
z
2
Cos[z]
-
3
x
2
z
3
Cos[z]
-4
2
x
2
z
Cos[z]Sin[z]+2
3
x
z
2
Cos[z]
Sin[z]+4
2
x
z
2
Sin[z]
-
3
x
Cos[z]
2
Sin[z]
+
3
x
z
3
Sin[z]
2
x
2
z
3
(-2z+xCos[z])
​
Submit
In[]:=
q3=QuestionObject["z is constrained by
2
w
+
3
x
+
4
z
=5 and z is constrained by log(w x)=z. What is the implicit derivative of
(w+x+z)

with respect to w?",AssessmentFunction[ImplicitD[
w+x+z

,{
2
w
+
3
x
+
5
z
==5,Log[wx]==z},{x,z},w]]]
Out[]=
z is constrained by
2
w
+
3
x
+
4
z
=5 and z is constrained by log(w x)=z. What is the implicit derivative of
(w+x+z)

with respect to w?
​
​
Submit
In[]:=
ImplicitD[
w+x+z

,{
2
w
+
3
x
+
5
z
==5,Log[wx]==z},{x,z},w]//TraditionalForm
Out[]//TraditionalForm=
-
w+x+z

(2
2
w
x+2
2
w
-3w
3
x
-5w
4
z
-3
3
x
+5x
4
z
)
w(3
3
x
+5
4
z
)
In[]:=
q3
Out[]=
z is constrained by
2
w
+
3
x
+
4
z
=5 and z is constrained by log(w x)=z. What is the implicit derivative of
(w+x+z)

with respect to w?
-
w+x+z

(2
2
w
+2
2
w
x-3
3
x
-3w
3
x
-5w
4
z
+5x
4
z
)
w(3
3
x
+5
4
z
)
​
Submit
In[]:=
q4=QuestionObject["z is bound by the implicit equation
t
z
==x. What is the derivative of sin(z) with respect to x and t?",AssessmentFunction[ImplicitD[Sin[z],
t
z
==x,z,x,t]]]
Out[]=
z is bound by the implicit equation
t
z
==x. What is the derivative of sin(z) with respect to x and t?
​
​
Submit
In[]:=
ImplicitD[Sin[z],
t
z
==x,z,x,t]//TraditionalForm
Out[]//TraditionalForm=
1-t
z
(-cos(z)+zlog(z)sin(z)-log(z)cos(z))
2
t
In[]:=
q4
Out[]=
z is bound by the implicit equation
t
z
==x. What is the derivative of sin(z) with respect to x and t?
1-t
z
(-Cos[z]-Cos[z]Log[z]+zLog[z]Sin[z])
2
t
​
Submit
In[]:=
q5=QuestionObject["What is the derivative of
3
z
-3xz=2 of z with respect to x?",AssessmentFunction[ImplicitD[
3
z
-3xz==2,z,x]]]
Out[]=
What is the derivative of
3
z
-3xz=2 of z with respect to x?
​
​
Submit
In[]:=
ImplicitD[
3
z
-3xz==2,z,x]//TraditionalForm
Out[]//TraditionalForm=
z
2
z
-x
In[]:=
q5
Out[]=
What is the derivative of
3
z
-3xz=2 of z with respect to x?
z
-x+
2
z
​
Submit
In[]:=
q6=QuestionObject["What is the derivative of sin(x z) with respect to x if z is implicitly defined by the polynomial equation
4
z
+2x=0?",AssessmentFunction[ImplicitD[Sin[xz],
4
z
+2x==0,z,x]]]
Out[]=
What is the derivative of sin(x z) with respect to x if z is implicitly defined by the polynomial equation
4
z
+2x=0?
​
​
Submit
In[]:=
ImplicitD[Sin[xz],
4
z
+2x==0,z,x]//TraditionalForm
Last but not least I can find the arity of a function:
I define a function for the arity of a function:
I remember seeing implicit derivatives in my calculus class so I think its cool that there's now a function to solve this.
Solve this problem:
Use Simplify to find the simplest form:
Solve this problem:
I use the following variable assignments, but its basically the same problem:
Solve an implicit differentiation problem with four variables: