Mastering the Chain Rule for Complex Derivatives

[nate808]

I understand hhow to use the chain rule for a simple 2 part composite function, however, I tend to have problems when it gets past that. Can someone please help me master these complex derivatives, or just a few quick tips would be nice

--Thanks

 

[BerkMath]

Are you talking about single variable functions such as: f_1(x), f_2(x), ...f_n(x), such that d(f_1(f_2(...(f_n(x)...)/dx=df_1/d_f2*df_2/d_f3*...*df_n/dx. Or are you talking about the sets of mulitvariable functions: (y_1y_2...y_n) such that y_i=f_i(u_1,u_2,...u_m), and (u_1,...u_m) such that u_i=h_i(x_1,...x_n)?

 

[nate808]

single variable

 

[mathwonk]

the number of compositions equals the number of factors in the derivative.

if you have three functions composed the derivatives is a product of three functions.

i.e. (fogoh)'(x) = f'(g(h(x)).g'(h(x)).h'(x).

 

[benorin]

Here is a verbal aid: Count the of 's.

For example,

$$\tan\left( \sec \left( \sqrt{x^{3}}\right) \right)$$

is read

"The tangent of the secant of the square root of the third power of x".

Each nested function is separated by an of,
this, with practice, makes the composition of functions pretty clear. Note that some functions need to be read in a non-standard way for this aid to work (e.g. rather than "x cubed," we read it "the third power of x.")

The derivative of the given function is what?

That is,

$$\frac{d}{dx}\tan\left( \sec \left( \sqrt{x^{3}}\right) \right) =?$$