How do you differentiate #y=sqrt(4+3x)#?

Answer 1

#y=3/(2(4+3x)^(1/2))#

Recall the formula for chain rule:

#color(blue)(bar(ul(|color(white)(a/a)dy/dx=dy/(du)(du)/dxcolor(white)(a/a)|)))# or #color(blue)(bar(ul(|color(white)(a/a)f'(x)=g'[h(x)]h'(x)color(white)(a/a)|)))#

and the formula for power rule:

#color(blue)(bar(ul(|color(white)(a/a)d/dx(x^n)=nx^(n-1)color(white)(a/a)|)))#
To start, recognize the inside and outside functions of #y=color(green)(sqrt(color(darkorange)(4+3x)))#.
Inside function: #y=color(darkorange)(4+3x)# Outside function: #y=color(green)(sqrt(a))#

How to Differentiate Using Chain Rule

#1#. Take the derivative of the outside function, #y=sqrt(a)#, but replace the #a# with the inside function, #4+3x#.
#2#. Multiply by the derivative of the inside function, #4+3x#.
Applying Chain Rule 1. The derivative of the outside function, #y=sqrt(a)#, would be #1/2a^(-1/2)#, using the power rule. However, #a# needs to be replaced by the inside function, so it becomes #1/2(4+3x)^(-1/2)#.
#y=sqrt(a)#
#color(red)(darr)#
#y=1/2a^(-1/2)#
#color(red)(darr)#
#y=1/2(4+3x)^(-1/2)#
2.#color(white)(i)#Then we need to multiply by the derivative of the inside function, #4+3x#, which becomes #3# using the power rule.
#y=1/2(4+3x)^(-1/2)#
#color(red)(darr)#
#y=1/2(4+3x)^(-1/2)(3)#
#color(green)(bar(ul(|color(white)(a/a)y=3/(2(4+3x)^(1/2))color(white)(a/a)|)))#
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Answer 2

To differentiate y = √(4 + 3x) with respect to x, you can use the chain rule.

  1. Start by rewriting the function as y = (4 + 3x)^(1/2).
  2. Then, differentiate the function term by term.
  3. The derivative of (4 + 3x)^(1/2) with respect to x is (1/2)(4 + 3x)^(-1/2) * (d/dx)(4 + 3x).
  4. Compute the derivative of 4 + 3x with respect to x, which is simply 3.
  5. Substitute this derivative back into the expression.
  6. Finally, simplify the result if possible.

So, the derivative of y = √(4 + 3x) with respect to x is: dy/dx = (1/2)(4 + 3x)^(-1/2) * 3 = 3/(2√(4 + 3x))

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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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