Chain rule understand concept??

[MathNoob94]

Hi, So I am learning chain rule and I've honestly become stumped and clueless as to how and when to use and combine the chain rule. Simple problems with one variable I can compute but two separate problems is where I get confused. Could someone please explain when I would use the chain rule combined with product rule etc. This question h(t) =(t+1)^2/3 (2t^2-1)^3

Looking at the solution manual it looks like complete gibberish to me.

Thank you!

 

[Ishuda]

Hi, So I am learning chain rule and I've honestly become stumped and clueless as to how and when to use and combine the chain rule. Simple problems with one variable I can compute but two separate problems is where I get confused. Could someone please explain when I would use the chain rule combined with product rule etc. This question h(t) =(t+1)^2/3 (2t^2-1)^3

Looking at the solution manual it looks like complete gibberish to me.

Thank you!

Product rules are for one function times another and chain rules are for one function of another. Sometimes it can work out that you can use either one but you come up with the same answer.
Product rule: (f g)' = f' g + f g'
Chain rule (f(g))' = f' g'

Suppose we have the function h(x)=(x³+1)². We take f(x)=x², f'(x)=2x, g(x)=x³+1, and g'=3x². To take the derivative we have
[h(x)]' = [(x³+1)²]' = [g²(x)]' = [f(g(x))]'
= f'(g(x)) g'(x)
= 2 g(x) g'(x) = 2 (x³+1) 3 x² = 6 x² (x³+1)
But we can also let f(x)=g(x)=x³+1 and get
[h(x)]' = [(x³+1)²]' = [f(x) g(x)]'
= f'(x) g(x) + f(x) g'(x) = 3x² (x³+1) + (x³+1)² 3x² = 6 x² (x³+1)

Note also that it doesn't matter how complicated the functions get, you can just keep working until the end. You have
h(t) =(t+1)^2/3 (2t²-1)³
and here you have the product rule AND the chain rule. To start with let f(t)=(t+1)^2/3 and g(t)=t³ and you have
h' = f' g + f g'
Now use the chain rule for g: g(k(t)) = k³(t) where k(t)=2t²-1 so
g' = 3 k² k'
and finally
h'(t) = [f(t) g(k(t))]' = f'(t) g(k(t)) + f(t) g'(k(t)) k'(t)

 

[Steven G]

Hi, So I am learning chain rule and I've honestly become stumped and clueless as to how and when to use and combine the chain rule. Simple problems with one variable I can compute but two separate problems is where I get confused. Could someone please explain when I would use the chain rule combined with product rule etc. This question h(t) =(t+1)^2/3 (2t^2-1)^3

Looking at the solution manual it looks like complete gibberish to me.

Thank you!

Personally I try never to use the chain rule. I use the general power rule instead. It says that if f(x) is a function to a power, say (g(x))^n, then f'(x) = n(g(x))^(n-1) * g'(x)
For example the derivative of h(t) =(t^2+11t)^(2/3)is h'(t) = (2/3)(t^2+11t)^(-1/3)* (2t+11)

 

[Steven G]

Hi, So I am learning chain rule and I've honestly become stumped and clueless as to how and when to use and combine the chain rule. Simple problems with one variable I can compute but two separate problems is where I get confused. Could someone please explain when I would use the chain rule combined with product rule etc. This question h(t) =(t+1)^2/3 (2t^2-1)^3

Looking at the solution manual it looks like complete gibberish to me.

Thank you!

If you insist on using the chain rule (your teacher says to use the chain rule) then you need to realize that your h(t) is a product of two other functions. Note the times symbol--h(t) =(t+1)^2/3 *(2t^2-1)^3. So you need to use the product rule to find h'(t) and you say that you can find the derivative of things like (t+1)^2/3 and (2t^2-1)^3. Then just use the product rule EXACTLY as it states and when the product rules tells you to take the derivative of one of the factors you need to understand that you may have to use the chain rule, product rule again, quotient rule or the general power rule to find that derivative.

Try to find the derivative of the function you posted and show us your work.

 

[pka]

Hi, So I am learning chain rule and I've honestly become stumped and clueless as to how and when to use and combine the chain rule. Simple problems with one variable I can compute but two separate problems is where I get confused. Could someone please explain when I would use the chain rule combined with product rule etc. This question h(t) =(t+1)^2/3 (2t^2-1)^3

Looking at the solution manual it looks like complete gibberish to me. I agree.

For many I taught basic calculus and then I taught graduate students who in turn would teach calculus. With both groups, I always found the same confusion about the chain rule. Oddly enough, it results not from calculus concepts but just basic algebra. That is, I found a pitiful understanding of the fundamental function composition.

Example: consider the function $\displaystyle y(x)=\sqrt{2-\sin^3(x^2+1)}$. Would you be surprised to know that asked to evaluate $\displaystyle y(3)$ with a calculator even some in-service teachers had trouble doing so?
The question is: where to begin? And that is the same question when asked to find $\displaystyle y'(x)$.

Let $\displaystyle a(x)=\sqrt{x~},~b(x)=2-x^3,~c(x)=x^2+1,~\&~d(x)=\sin(x)~.$
The function $\displaystyle y$ is a composition of those four functions. But what is the order?
Is it $\displaystyle y(x)=a\circ b\circ d\circ c(x)=a( b( d( c(x))))~\Large ?$
To evaluate $\displaystyle y(3)$ we first square three, add one, find the sine of nine, cube that value, subtract that result from two and finally find the square root.

The exact same ideas apply to the CHAIN RULE.

$\displaystyle y'(x)=[a'( b( d( c(x))))]\cdot [ b'( d( c(x)))]\cdot [d'( c(x))]\cdot [c'(x)]$

So
$\displaystyle y'(x) = \underbrace {\left[ {\dfrac{1}{{2\sqrt {2 - {{\sin }^3}\left( {{x^2} + 1} \right)} }}} \right]}_{a'(b(d(c(x))))} \cdot \underbrace {\left[ { - 3{{\sin }^2}\left( {{x^2} + 1} \right)} \right]}_{b'(d(c(x)))} \cdot \underbrace {\left[ {\cos \left( {{x^2} + 1} \right)} \right]}_{d'(c(x))} \cdot \underbrace {\left[ {2x} \right]}_{c'(x)}$

I have no doubt you may not have done all those derivative forms. But the point is the format of the chain rule.

 

[stapel]

I've honestly become stumped and clueless as to how and when to use and combine the chain rule.

I'm not sure what you mean by "combining" the Chain Rule...? "Combining" how? With what?

As for "when", you use it any time you have layers of functions, like a sine that's inside a cube that's inside a square root. In other words, any time you have a situation which can be restated as the composition of two or more functions, you apply the Chain Rule when differentiation.

When evaluating composed functions, you start from the inside, plugging the value into the variable at the inner-most position, and then go outwards. For instance, in the example above, you'd first have to evaluate the sine, then plug the resulting value into the cube, and then plug that result into the square root.

When differentiating composed functions, you start from the outside, and work your way inward, differentiating the layers as you go. In the example above, you'd first differentiate the square root (or, which is the same thing, the one-half power), leaving the insides of the square root (the cube of the sine) untouched; then you'd add a "times" symbol, and differentiate the cube, leaving the insides (the sine) untouched; then you'd add another "times" symbol, and differentiate the sine.

For other practical (that is, non-technical) discussions of this topic, please try the following links:

. . . . ."Peel the onion"

. . . . ."(Blob)"