Using 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

Correct. Each of the functions in I and III is a composition of component functions. In I, let f(x)=sinxf(x)=sin⁡x and let g(x)=3x2g(x)=3×2. Then sin(3×2)=f(g(x))sin(3×2)=f(g(x)), and the chain rule would give ddxsin(3×2)=f′(g(x))g′(x)=cos(3×2)6xddxsin(3×2)=f′(g(x))g′(x)=cos(3×2)6x. In III, let f(x)=1x=x−1f(x)=1x=x−1 and g(x)=8×4−2xg(x)=8×4−2x. Then 18×4−2x=f(g(x))18×4−2x=f(g(x)), and the chain rule would give ddx18x4−2x=f′(g(x))g′(x)=−1(8×4−2x)2⋅(32×3−2)ddx18x4−2x=f′(g(x))g′(x)=−1(8×4−2x)2⋅(32×3−2). In II, the function y=extanxy=extan⁡x is the product of the exponential function and the tangent function and is not the composition of two component functions. The product rule would be used to find the derivative, and the chain rule would not be appropriate.

The chain rule is an appropriate method to find the derivative of a function which is a composition of two or more functions. The general statement of the chain rule is:

d[f(g(x))]/dx = f'(g(x)) * g'(x)

Using this rule, we can find the derivative of a function that involves nested functions.

Now, let’s look at the given functions and determine which ones involve a composition of functions.

y = sin(3x^2)
In this function, we have the outer function as sin(x) and the inner function as 3x^2. Therefore, this function can be solved using the chain rule.

To find the derivative, we can use the following steps:

Let u = 3x^2
y = sin(u)

Now, we can find the derivatives of the functions

dy/du = cos(u)
du/dx = 6x

Using the chain rule, we can find the derivative of y with respect to x:

dy/dx = dy/du * du/dx
= cos(u) * 6x
= 6x cos(3x^2)

Therefore, the function y = sin(3x^2) can be solved using the chain rule.

y = ex*tan(x)
In this function, we have two functions, but they are multiplied by each other, not nested. Hence, the chain rule cannot be used here to find the derivative.

y = 18x^4 – 2x
In this function, we have a polynomial function and a constant. Therefore, the chain rule is not applicable here because there is no composition of functions.

In summary, the chain rule is an appropriate method to find the derivative of the function y = sin(3x^2) because it involves a composition of functions. However, the chain rule is not applicable for the other given functions.

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