Applying the Chain Rule to Find Derivatives of Composite Functions

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

The chain rule is a method used to find the derivative of a composite function

The chain rule is a method used to find the derivative of a composite function. It allows us to differentiate functions that are composed of multiple nested functions. The chain rule states that if we have a function f(g(x)), the derivative of this function with respect to x can be found by multiplying the derivative of the outer function (f'(g(x))) by the derivative of the inner function (g'(x)).

Let’s analyze each of the given functions to determine if the chain rule applies:

1. y = sin(3x^2):
This function is a composition of two functions: the outer function is sin(x) and the inner function is 3x^2. Since both of these functions are differentiable, we can use the chain rule to find the derivative. The derivative of the outer function sin(x) is cos(x), and the derivative of the inner function 3x^2 is 6x. Applying the chain rule, the derivative is given by:
dy/dx = cos(3x^2) * 6x = 6x * cos(3x^2)

2. y = e^tan(x):
This function is a composition of two functions: the outer function is e^x (where x is replaced by tan(x)) and the inner function is tan(x). Since both of these functions are differentiable, we can use the chain rule. The derivative of the outer function e^x is e^x, and the derivative of the inner function tan(x) is sec^2(x). Applying the chain rule, the derivative is given by:
dy/dx = e^tan(x) * sec^2(x)

3. y = 18x^4 – 2x:
This function consists of two terms being added together – 18x^4 and -2x. Each of these terms is an elementary function (polynomial), and elementary functions are differentiable. Thus, there is no nested function composition, so the chain rule is not needed to find the derivative. The derivative of 18x^4 is 72x^3, and the derivative of -2x is -2. The derivative of the given function is:
dy/dx = 72x^3 – 2

In summary, the chain rule is applicable for the functions y = sin(3x^2) and y = e^tan(x), but not for y = 18x^4 – 2x.

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