Determining Functions for Chain Rule Differentiation | Sinusoidal Composition and Polynomial

For which of the following functions is the chain rule an appropriate method to find the derivative with respect to x ?y=sin(3×2)y=extanxy=18×4−2x

To determine which of the following functions can be differentiated using the chain rule, we need to check if they involve compositions of functions

To determine which of the following functions can be differentiated using the chain rule, we need to check if they involve compositions of functions. The chain rule is applicable when a function can be written in the form y = f(g(x)), where f and g are both functions.

1. y = sin(3x^2):
In this case, the function involves the composition sin(g(x^2)), which means the chain rule is applicable.

To apply the chain rule, we differentiate the outer function (sin) first, and then multiply it by the derivative of the inner function (3x^2):
dy/dx = cos(3x^2) * d(3x^2)/dx = cos(3x^2) * 6x = 6x cos(3x^2)

2. y = ex * tan(x):
This function involves the product of two functions (ex and tan(x)), but not a composition of functions. Therefore, the chain rule is not applicable in this case.

To differentiate this function, we can use the product rule:
dy/dx = ex * sec^2(x) + tan(x) * d(ex)/dx = ex * sec^2(x) + ex * tan(x)

3. y = 18x^4 – 2x:
This function is a polynomial, and the chain rule is not needed for polynomial functions. We can differentiate it directly using the power rule:
dy/dx = 4 * 18x^3 – 1 = 72x^3 – 1

So, the only function where the chain rule is applicable is y = sin(3x^2).

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