Chain Rule Reversal
Chain Rule Reversal - Related Videos
01:30
The Chain Rule
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2026
A system of interconnected gears provides a concrete physical interpretation of the Chain Rule in calculus. Consider three gears arranged in sequence, where the rotational speeds of the first, second, and third gears are represented by the variables x, z, and y, respectively. The first gear drives the second, and the second drives the third, so the motion of each gear depends on the one preceding it. This structure naturally leads to a two-stage variable relationship that can be analyzed using...
01:23
The Chain Rule: Problem Solving
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2026
The thermal expansion of a metal rod shows the application of the Chain Rule when one physical quantity depends on another that varies with time. As the rod is heated, its length changes according to linear thermal expansion, while the temperature of the system varies quadratically with time.For linear thermal expansion, the length L of the rod depends on temperature T such that the rate of change of length with respect to temperature is constant:where L0 = 2 m is the initial length of the rod,...
01:29
Multivariable Chain Rule
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2026
When a variable z depends on two intermediate variables, x and y, and both x and y vary with respect to a third variable t, the dependence of z on t is indirect. Although t does not explicitly appear in the expression for z, any change in t produces corresponding changes in x and y, which in turn alter the value of z. The objective is to determine the total derivative of z with respect to t, denoted as dz/dt.Assuming that all functions involved are differentiable, the total change in z can be...
01:34
Woodward–Hoffmann Selection Rules and Microscopic Reversibility
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2023
Electrocyclic reactions, cycloadditions, and sigmatropic rearrangements are concerted pericyclic reactions that proceed via a cyclic transition state. These reactions are stereospecific and regioselective. The stereochemistry of the products depends on the symmetry characteristics of the interacting orbitals and the reaction conditions. Accordingly, pericyclic reactions are classified as either symmetry-allowed or symmetry-forbidden. Woodward and Hoffmann presented the selection criteria for...
02:27
Measuring Hepatitis C Viral Load Using Quantitative Reverse Transcription Polymerase Chain Reaction
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2026
Source: Ren, S., et al. A Protocol for Analyzing Hepatitis C Virus Replication. J. Vis. Exp. (2014).This video demonstrates the procedure for quantifying Hepatitis C viral RNA using quantitative reverse transcription PCR to measure viral load in infected cell cultures.
FAQs
Look for a composite expression whose outer operation is applied to an inner function, then check whether the inner function’s derivative also appears in the integrand. For example, a power, exponential, logarithmic, or trigonometric expression may contain the candidate inner function. The derivative match indicates that substitution can simplify the calculation.
The derivative match supplies the differential needed for the substitution. When u = g(x) and du = g′(x) dx, the corresponding part of the integrand can be rewritten in terms of u and du. This removes the original inner expression from the calculation and exposes a simpler integral associated with the outer function.
The strategy is especially useful when an integrand contains a recognizable composite structure involving products, powers, exponentials, logarithms, or trigonometric functions. Suitability depends on whether the derivative of the selected inner expression is also present. When that relationship appears, substitution can reduce a complicated-looking expression to a more direct form.
Choose the inner function and label it u = g(x). Differentiate this choice to obtain du = g′(x) dx, then replace the matching expression in the integral with du and rewrite the remaining terms using u. Integrate the simplified expression, and finally replace u with g(x) to express the result in the original variable.
Set the repeated inner expression equal to u rather than treating the entire integrand as one complicated object. Its derivative provides du, while the outer power or exponential form becomes an expression in u. After integration, replacing u with the original inner function restores the connection to the initial variable and composite expression.
This strategy connects integration directly with the chain rule used in differentiation. Practicing it helps students recognize how composite functions are built and dismantled, while also preparing them for more advanced integration techniques. In mathematical modeling and analysis, that connection supports the evaluation of integrals involving composite expressions and their resulting relationships.