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# Applying the chain rule graphically 3 (old)

Sal solves an old problem where the graphs of functions f and g are given, and he evaluate the derivative of [g(f(x))]² at a point. Created by Sal Khan.

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• No teacher or professor has ever been able to explain this to me! I've always heard that you can't treat differentials like numbers, but it always seems to work out that way (ex./ Derivation of Euler's Formula, the chain rule, implicit differentiation, etc.). What's the deal with that? Why shouldn't we think of them as "fractions" if we always end up treating them that way?
• You can't treat derivatives as true fractions because some of the terms are fixed and some of them are variable and you risk mixing them up. For example the derivative of log base b of x is the derivative of ln(x)/ln(b) (from logarithm properties, change of base formula). If you don't treat this as a derivative but as a fraction, you obtain the result b/x, because you'll take each derivative separately - as in deriv. of ln(x)=1/x and deriv. of ln(b)=1/b; (1/x)/(1/b)=b/x, which is wrong. Looking at Sal's example here (https://www.khanacademy.org/math/differential-calculus/taking-derivatives/chain_rule/v/derivative-of-log-with-arbitrary-base) you'll see that he takes out of the derivative the 1/ln(b), which is a constant term and the result is totally different, 1/(ln(b))x. The point here is that the ln(x) will vary with x, while the ln(b) has one and only one value.

Also to treat them as fractions would mean to miss a part of the derivative. For example, the deriv of e^cos(x), treated as a simple fraction would be just e^cos(x) because you would simplify the second part, which is incorrect. This derivative is e^cos(x) multiplied by the derivative of cos(x). Ths second part would be eliminated by simplification if you would treat derivatives as fractions.
• for along time i have been facing a problem in understanding the term something respect to something
if anybody can help I'll be pleased
• The derivative of y(or any other variable) with respect to x(or any other variable) is the rate of change of the value of the variable 'y' with a extremely small change in the value of the variable 'x' or dy/dx
The rate of change of distance with reference to time is called distance.
The rate of change of y can be found with any other variable other than x such as a which will represent the change in 'y' with a change in 'a'
• In which scenario would the derivative of a function be its negative?
• The simplest case: the zero function. If ƒ(x) = 0 for every x, then clearly ƒ '(x) = 0 = -ƒ(x).
• Can someone explain why the derivative of (sin(3x))^2 = 3sin(6x)?
• Not trying to be mean with redthumb, but he did a small mistake when he put that:
6•sin(2u) = 2•sin(u)•cos(u)
Just by comparing to the double-angle identity formula right above it we can see that the equation is wrong, thus leading to the incorrect result:
[3•sin(6•x)] = sin(3•x)•cos(3•x)
which is just the same as stating that:
sin(6•x) = (1 / 3)•sin(3•x)•cos(3•x)
which contradicts the double-angle identity formula.
Now what I'd do to apply that formula and get to the 3•sin(6x) answer is, after finding that:
d/dx(sin^2(3x)) = 6sin(3x)cos(3x)
sin(2θ) = 2sin(θ)cos(θ)
3x = θ
So now we have as a result:
6sin(θ)cos(θ)
Manipulating the DAI formula we get:
sin(θ)cos(θ) = sin(2θ) / 2
Substituting this in our result we have:
6sin(θ)cos(θ) = 6(sin(2θ) / 2) = (6 / 2)sin(2θ) = 3sin(2θ)
θ = 3x, so finally:
3sin(2•3x) = 3sin(6x)
Summarizing:
d/dx[sin²(3x)] = 6sin(3x)cos(3x) = 6[sin(2•3x) / 2] = 3sin(6x)
(1 vote)
• What would g-prime of 5 have been? Since the slope of the line changes at that exact point.
• The derivative does not exist at that point.
There is no tangent line at that point.
The slope on the negative side of 5 does not equal the slope on the positive side of 5.
• what about the exponent 2 of the G(x)?? i think it won't lead the G'(5) to 0.
• @ you can see the fully expanded application of the chain rule on G(x). G'(x)= a product, and since one of the factors equals 0, the whole thing surely equals 0. Other than that, if the input on G(x) is constant, squaring it gives back a constant as well. When we graph G(x)=C on some interval, the slope of it is 0, and thus G'(x)=0 as well.
(1 vote)
• What is the physical meaning of differentiating a function with respect to other function and how can we physically feel it and interpret graphically?
Like we can physically feel that what does differentiating a function with respect to x means (like we can graphically interpret; first principle of derivative).
Thank You!
(1 vote)
• It's like peeling layers from an onion. You start with a degree 3 (cubic) onion and peel off the first layer in such a way that inside you find a second degree (square) onion. From the 3 dimensional onion you started with, after peeling off the first layer, you will obtain a bi-dimensional onion (the photo of the onion). Derivatives take things "one level down"; integrals, which are inverses of derivatives, take things "one level up". So if you start with the photo of an onion, by integrating it, you'll find the whole onion.
There is a difference between derivatives and differentials. Derivatives are taken for one variable while differentials are taken (somewhat circularly) with respect to all the variables. The derivative of an onion with respect to one variable will be the photo of the onion from one side. The differential of the onion can generate 2 views, one from side and one from (for example) the top of the onion. Of course, this can be generalized to an arbitrary number of dimensions, each with its own variable. The higher the degree of the onion (the higher dimensional the onion is), the more views you can generate by differentiating it.
• without using the triple derivative, would this process be right?
using the power rule and the chain rule at the same time,, basically.
G'(x)= 2(g(f(x))) * g'(f(x))* f'(x) ?
• You can use whatever symbol you want to designate your functions. So for starters you have all the letters in the alphabet (excluding those you design as variables), if you need more you can use greek letters, and if you need still more you can either decide to designate functions by a combination of letter and numbers (for example `f₁(x)`, `f₂(x)`, `f₃(x)`) or use abstract symbols or drawings to represent the functions.