If f ( x ) = x + 2 f\left(x\right)=\sqrt{x+2} f ( x ) = x + 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> , find the differential d y dy d y when x = 2 x=2 x = 2 and d x = 0.01 dx = 0.01 d x = 0.01 .

Let's try this problem out. So here, we're told if f of x is equal to the square root of x plus 2, find the differential dy when x is equal to 2, and dx is equal to 0.01. Okay. Now to solve this problem, we need to recognize that the differential dy, which is what we're trying to solve for, is equal to the derivative of our function times dx. And the way that we can find these values is by first figuring out what the derivative of our function is going to be. Now the derivative of our function is gonna be the derivative of the square root of x plus 2. And the square root of x plus 2 can be rewritten as x plus 2 to the one half power. Now that's going to allow me to take this derivative more easily since I can clearly use the power rule right here. So doing this, we're going to get 1 half times x plus 2 raised to the negative one half power. Now typically, you need to multiply this by the derivative of the inside since we would use the chain rule. But notice the derivative of x here is just gonna be 1, and then the derivative of 2 is 0. So this just being multiplied by 1, and so we don't even have to worry about that. This right here would be our derivative. Now what I'm going to do from here is I'm going to simplify this because one half times x plus 2 to the negative one half power can also be written as 1 over 2 times the square root of x plus 2. So this is f prime of x. And if I go ahead and rewrite this differential, we'll get 1 over 2 times the square root of x plus 2, all multiplied by the x. And I can go ahead and take my values up here, x equals 2 and d x equals point 1, and I can plug them into this function. So we're gonna get 1 over 2 times the square root, and then this case, x is 2 plus 2, then that's multiplied by d x point 1. Now from here, we can go ahead and simplify. So this whole denominator here, that's all going to become 1 fourth, and that's all gonna be multiplied by 0.01. Now if you go ahead and do this math right here, the whole thing should come out to 0.0025. So it's a very small number, but this right here would be our solution for d y. So that is how you can find dy, this differential in this equation, and that is how you can solve this problem. Hope you found this video helpful, and let's move on and get some more practice.

5. Applications of Derivatives

Differentials

Business Calculus

5. Applications of Derivatives

Differentials

Struggling with Business Calculus?

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Multiple Choice

If $f\left(x\right)=\sqrt{x+2}$ , find the differential $dy$ when $x=2$ and $dx = 0.01$ .

0.02

0.0025

2.00

0.025

Verified step by step guidance

Step 1: Start by identifying the given function f(x) = √(x + 2). The goal is to find the differential dy when x = 2 and dx = 0.01.

Step 2: Recall the formula for the differential dy, which is given by dy = f'(x) * dx. This means we need to compute the derivative of f(x), denoted as f'(x).

Step 3: Differentiate f(x) = √(x + 2) using the chain rule. The derivative of √(x + 2) is (1/2) * (x + 2)^(-1/2) * (d/dx of (x + 2)), which simplifies to (1/2) * (x + 2)^(-1/2).

Step 4: Substitute x = 2 into the derivative f'(x). This gives f'(2) = (1/2) * (2 + 2)^(-1/2) = (1/2) * (4)^(-1/2).

Step 5: Use the formula dy = f'(x) * dx. Substitute f'(2) and dx = 0.01 into the equation to compute dy. This gives dy = [(1/2) * (4)^(-1/2)] * 0.01.

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Struggling with Business Calculus?

If f(x)=x+2f\left(x\right)=\sqrt{x+2}f(x)=x+2, find the differential dydydy when x=2x=2x=2 and dx=0.01dx = 0.01dx=0.01.

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Differentials practice set

If $f\left(x\right)=\sqrt{x+2}$ , find the differential $dy$ when $x=2$ and $dx = 0.01$ .