How do you use the chain rule to differentiate #root3(4x+9)#?

Answer 1

#4/3 (4x+9)^(-2/3)#

Consider 4x+9 as some function p of x. Thus #f=p^(1/3)#

Now differentiate f with respect to p and then p with respect to x, to get differential of f w.r.t x. This is the chain rule. It works out as follows:

#(df)/dx= (df)/(dp) * (dp)/dx#
=# 1/3 p^(-2/3) * d/dx (4x+9)#
=#1/3 (4x+9)^(-2/3) * (4)#
=#4/3 (4x+9)^(-2/3)#
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Answer 2

To differentiate √(3(4x + 9)) using the chain rule, first identify the outer function (the square root) and the inner function (3(4x + 9)). Then, differentiate the outer function with respect to the inner function, and multiply the result by the derivative of the inner function.

The derivative of √u, where u is a function of x, is (1/2) * (u^(-1/2)) * du/dx.

For this problem:

Outer function: √u Inner function: 3(4x + 9)

Derivative of the outer function: (1/2) * (u^(-1/2)) * du/dx Derivative of the inner function: d(3(4x + 9))/dx

Now, calculate the derivatives:

Derivative of the outer function: (1/2) * (3(4x + 9))^(-1/2) * d(3(4x + 9))/dx Derivative of the inner function: 12

Finally, multiply the two derivatives:

(1/2) * (3(4x + 9))^(-1/2) * 12

This simplifies to:

6/(√(3(4x + 9)))

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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