Microsoft Math 3.0: Solving Complex Equations

[ARYT]

Homework Statement

Homework Equations

Answers by Microsoft Math 3.0

The Attempt at a Solution

This is confusing. Too many parentheses. We used to solve a composite of two or three functions.

 

[Vagrant]

If f(x) = h(g(x)) then f'(x) = h'(g(x)) × g'(x).
In words: differentiate the ‘outside’ function, and then multiply by the derivative of the
‘inside’ function.

Let's take an example,
y= sin (x²)
dy/dx= d(sin x²)/dx *d(x²)/dx

Treat all the parentheses in a similar manner.

 

[ARYT]

I know, I could solve the third one myself, but the first one :(

 

[Vagrant]

Break up the expression as:
z(x) = (1+v(x))⁵
v(x) = (2-u(x))³
u(x) = (6+7x²)⁹

finally you have y = 10*z(x)

now dy/dx = dy/dz *dz/dx

 

[ARYT]

(fog(x) )'=g' (x) f' (g(x) )

we have this general rule for two functions only. Give me sth for n functions.

 

[cristo]

(fog(x) )'=g' (x) f' (g(x) )

we have this general rule for two functions only. Give me sth for n functions.

shramana has given you a pretty good hint. How about you try to follow it? Post your work, and we will point out any errors.

 

[Vagrant]

Now let's say
g(x)= g(h(x))
so you'll have g'(x)=h'(x)*g'(h(x)) [using f'(x)=g' (x) f' (g(x) )]

substitute the value of g(x) in f'(x) and so on...

 

[HallsofIvy]

Do it step by step: If y= 10u⁵, the dy/dx= 50u⁴ du/dx.

And u= 1+v³ so du/dx= 3v^2 dv/dx.

v= ...?

 

[ARYT]

OK. Here I've tried to solve the first one.

 

[ARYT]

Although it's too long. I won't be able to do it without a software (for multiplication and things like that). Also, We can't use calculator.

 

[ARYT]

And the second one which compare to the answer given by the Microsoft Math is wrong:

 

[Vagrant]

Although it's too long. I won't be able to do it without a software (for multiplication and things like that). Also, We can't use calculator.

Not at all. You just have to differentiate the functions in the parentheses as you move inwards.

f'(x)=10*z'(x)
z'(x)=5*(1+v(x))⁴*v'(x)
v'(x)=3*(1-u(x))²*(-u'(x))
u'(x)=9*7*d(x⁴)/dx

Now substitute the values of u'(x), v'(x),z'(x) and u(x), v(x) in f'(x).

 

[Vagrant]

And the second one which compare to the answer given by the Microsoft Math is wrong:

Step 2 is wrong.
v'(x) = d(ln(ln sec x))/dx