chain rule with symbolic functions

I define the following symbolic variables:

syms s k mu2(s,k) mu3(s,k) mu4(s,k) 
>> mu2(s, k) = 1 + 6*k^2 - 24*k*s^2 + 25*s^4;
>> mu3(s,k) = 108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s;
>> mu4(s,k) = 3 + 3348*k^4 - 28080*s^2*k^3 + 1296*k^3 - 6048*s^2*k^2 ...
+ 252*k^2 - 123720*s^6*k + 8136*s^4*k - 504*s^2*k ...
+ 24*k + 64995*s^8 - 2400*s^6 - 42*s^4 + 88380*k^2*s^4;
>> 
>> Scap = mu3(s,k)/mu2(s,k)^3/2

Scap =(108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s)/(2*(6*k^2 - 24*k*s^2 + 25*s^4 + 1)^3)

When I take the 1st derivative of Scap with respect to s I would expect MatLab to apply the chain rule as Scap is not directly a function of s. Instead I get the following:

>> diff(Scap,s)
ans =
(108*k^2 - 1404*k*s^2 + 36*k + 2550*s^4 - 228*s^2 + 6)/(2*(6*k^2 - 24*k*s^2 + 25*s^4 + 1)^3) + (3*(- 100*s^3 + 48*k*s)*(108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s))/(2*(6*k^2 - 24*k*s^2 + 25*s^4 + 1)^4)

How can I have MatLab explicitly apply the chain rule ?

Thank you so much

mauede

[mauede]

My email is: [e-mail]

Answer 1

[Walter Roberson]

The definition

Scap = mu3(s,k)/mu2(s,k)^3/2

is procedural not symbolic. The current values of mu3(s,k) and mu2(s,k) are fetched and substituted, giving the expression you see, which is a direct function of s that has no "memory" of being formed by mu3 and mu2 .

If you want explicit chain rule in terms of mu2 and mu3, then you need to define Scap first in terms of mu2 and mu3 and do the differentiation before you give the formula for mu2 and mu3.

syms s k mu2(s,k) mu3(s,k) mu4(s,k) Scap(s,k)
Scap = mu3(s,k)/mu2(s,k)^3/2;
dScap = diff(Scap(s,k),s)          %explicit chain rule result
mu2(s, k) = 1 + 6*k^2 - 24*k*s^2 + 25*s^4;
mu3(s,k) = 108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s;
subs(dScap)                        %substitute the defined mu2 and mu3 into the chain rule

Answer 2

[asma ahmed]

How to answer this question ؟

Recall the Chain Rule for differentiation: GIven a function of the form y = f(g(x)), the derivative is:

y ' = f '(g(x)) * g'(x)

Using the formula above as outlined in the Solution Template, work out the derivative of y = sin^3( x ) (NOTE: This can also be written as (sin x)^3 ). Then check your answer by using the diff command directly on f(g(x)).

syms ;  % NOTE below that you have TWO symbolic variables to define this time!
f=@(u) ;  % Write f (outer function) as a function of u to minimize confusion
g=@(x) ;  % Write g (inner function) as a function of x
df=diff( );  % derivative of f(u)
df_at_g=subs( );  % replace u with g(x) 
dg=diff( );  % derivative of g(x)
dy=   % Multiply the appropriate parts above for the final answer
dyalt=diff( )  % Differentiate f(g(x)) directly to compare answers

Answer 3

[mauede]

I tried Walter's technique. It does not work for me. See the following attempts. Thank you in advance for your help [e-mail]

syms s k mu2(s,k) mu3(s,k) mu4(s,k) Scap(s,k) Kcap(s,k) ...
     J J1 J2 J3 J4 J5 J6 ...
     dsmu2 dsmu3 dsmu4 dkmu2 dkmu3 dkmu4 

%--------- define Scap as a function of (mu2, mu3)
Scap = mu3/mu2^(3/2)
Scap(s, k) =
mu3(s, k)/mu2(s, k)^(3/2)
>> dsScap = diff(Scap,s)
dsScap(s, k) =
diff(mu3(s, k), s)/mu2(s, k)^(3/2) - (3*diff(mu2(s, k), s)*mu3(s, k))/(2*mu2(s, k)^(5/2))
>> mu2(s,k) = 1 + 6*k^2 - 24*k*s^2 + 25*s^4
mu2(s, k) =
6*k^2 - 24*k*s^2 + 25*s^4 + 1
>> mu3(s,k) = 6*s - 76*s^3 + 510*s^5 + 36*s*k - 468*k*s^3 + 108*s*k^2
mu3(s, k) =
108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s
>> subs(dsScap)
ans(s, k) =
diff(mu3(s, k), s)/mu2(s, k)^(3/2) - (3*diff(mu2(s, k), s)*mu3(s, k))/(2*mu2(s, k)^(5/2))
>> subs(dsScap,mu2)
ans(s, k) =
subs(diff(mu3(s, k), s), s, 6*k^2 - 24*k*s^2 + 25*s^4 + 1)/mu2(6*k^2 - 24*k*s^2 + 25*s^4 + 1, k)^(3/2) - (3*subs(diff(mu2(s, k), s), s, 6*k^2 - 24*k*s^2 + 25*s^4 + 1)*mu3(6*k^2 - 24*k*s^2 + 25*s^4 + 1, k))/(2*mu2(6*k^2 - 24*k*s^2 + 25*s^4 + 1, k)^(5/2))
>> subs(dsScap,mu3)
ans(s, k) =
subs(diff(mu3(s, k), s), s, 108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s)/mu2(108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s, k)^(3/2) - (3*mu3(108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s, k)*subs(diff(mu2(s, k), s), s, 108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s))/(2*mu2(108*k^2*s - 468*k*s^3 + 36*k*s + 510*s^5 - 76*s^3 + 6*s, k)^(5/2))

[Walter Roberson]

See http://uk.mathworks.com/matlabcentral/answers/255123-subs-does-not-substitutes-in-the-derivation-maura-monville-gmail-com