<p>Our class is learning the Chain Rule and Implicit Differentiation. I get how to use the Chain Rule but I do not get why they are used that way. Does the Chain Rule basically apply for any equation except that the derivative of the "inside function" is equal to 1 so we don't write it out?</p>
<p>Basically, the chain rule always applies. In case like sin(x) the derivative is cos(x), but in reality it’s cos(x)*derivative of x, which is the same thing as 1 so you just don’t write it out.</p>
<p>The Chain Rule makes much more sense if you understand the idea of composition of functions. If you understand composition of functions, then bsaically you’re taking the derivative of each interior level until you get to the end.</p>
<p>Example 1 (Basic)
So, for instance, let f(x) = (3x^2 - 5)^3.
We can define f(x) as the composition of g(h(x)), where h(x) = 3x^2 - 5 and g(x) = x^3. Now normally, we can accept that g’(x) = 3x^2 and h’(x) = 6x. So d/dx[g(h(x))] = g’(h(x))*h’(x) = 3(3x^2-5)^2 * 6x = 18x(3x^2 - 5)^2.</p>
<p>Example 2 (Advanced)
Examine the function k(x) = sin^2 (4x-5)^2 = [sin ((4x-5)^2)]^2.
This is really the composition of four functions f(x) = sin x, g(x) = x^2, h(x) = 4x-5, and j(x) = x^2. Then k(x) = g(f(j(h(x)))). [It’s the square of the sin of the square of (4x-5).] k’(x) = g’(f(j(h(x))) * f ‘(j(h(x)) * j’(h(x)) * h’(x).</p>
<p>Now normally, g’(x) = 2x, so g’(f(j(h(x))) = 2(sin (4x-5)^2)
f ‘(x) = cos x, so f ‘(j(h(x)) = cos (4x-5)^2
j’(x) = 2x, so j’(h(x)) = 2(4x-5)
h’(x) = 4.</p>
<p>So k’(x) = 2(sin (4x-5)^2)(cos (4x-5)^2)<em>2(4x-5)</em>4 = (64x - 80)(sin (4x-5)^2)(cos (4x-5)^2).</p>
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<p>Hope that helps.</p>
<p>I have a really short straightfoward confusion on related rates. For cone problems where water is pouring in and, at the same time, leaking out… do we simply subtract the two to find the NET rate of the the increase/decrease in volume of the water?</p>
<p>Please reply fast, I’m studying related rates for 2 more hours, then I’m going to watch the Iowa vs. Penn State football game.</p>
<p>In essence, yes. The rate of the water entering the cone is positive, and the rate of the water leaving the cone is negative. The net is positive when water is entering the cone faster than it is leaving, and negative when the reverse is true.</p>
<p>Thank you very much TheMathProf, your explanations helped me understand much more deeply!</p>
<p>Glad it helped.</p>
<p>My students don’t tend to understand composition of functions before entering my class, and so they struggle a lot with the idea of the Chain Rule. They basically get to asking, “When do I stop taking the derivative?” :)</p>
<p>Ah, the perils of attempting to teach so many fundamental concepts in such a short span of time … </p>
<p>If the syllabus started earlier, students should ideally start graphing composite functions manually before they get to the chain rule, and watching what happens to the slope as they gradually change the scalar multiple of an internal function inside an exterior one.</p>
<p>The FDWK book actually does this to an extent with trig functions, introducing their derivatives (3.5) prior to the chain rule (3.6).</p>
<p>Of course, what should really happen is that composition of functions should be well covered in the precalculus course, and ideally, only a cursory review would be necessary in AP Calc. It’s covered in my school’s precalculus course, but not as well as it should be. Unfortunately, I have no control over our school’s precalculus curriculum, so we have to compensate as well as we can at the AP level (where I surprisingly have more control).</p>