If f ( x ) = x 3 − 8 x + 6 f\left(x\right)=x^3-8x+6 f ( x ) = x 3 − 8 x + 6 , find the differential d y dy d y when x = 2 and d x = 0.2 dx = 0.2 d x = 0.2 .

this problem, we are given this function, f of x is equal to x cubed minus 8x plus 6, and we're asked to find the differential dy when x is equal to 2 and dx is equal to 0.2. Now we learned to find the differential dy, it's going to be the derivative, f prime of x times d x. Now, our my the first thing I'm gonna do is find the derivative of our function right here. So I can do that by using the power rule. So doing this, we're going to have the derivative of x cubed, which is 3x squared. We're going to have the derivative of negative 8x, which is negative 8. And then this 6 is gonna go away, so this is all we're gonna be left with for the derivative. So we're going to have the derivative 3x squared minus 8. That's going to be multiplied by the x. Now what we can do is plug in the values that we're given. We can see here that x is equal to 2. So we'll have 3 times 2 squared minus 8, then that's going to be multiplied by dx, which is 0.2. So at this point, all you need to do is multiply these values together. You can subtract these and then multiply them. What you should get is that this whole thing comes out to 0.8, because this all is going to evaluate the 4, and then you multiply that by 0.2, you get 0.8. So this right here is going to be the solution for d y, and that is the answer to this problem. Hope you found this video helpful.

4. Applications of Derivatives

Differentials

Calculus

4. Applications of Derivatives

Differentials

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Multiple Choice

If $f\left(x\right)=x^3-8x+6$ , find the differential $dy$ when x = 2 and $dx = 0.2$ .

0.8

0.4

3.2

-1.3

Verified step by step guidance

First, understand that the differential dy represents the change in the function f(x) when x changes by a small amount dx. We need to find dy when x = 2 and dx = 0.2.

To find dy, we start by calculating the derivative of the function f(x) = x^3 - 8x + 6. The derivative, f'(x), gives us the rate of change of the function with respect to x.

Differentiate f(x) = x^3 - 8x + 6 with respect to x. The derivative f'(x) is found using the power rule: f'(x) = 3x^2 - 8.

Evaluate the derivative at x = 2. Substitute x = 2 into f'(x) to find f'(2). This gives us the rate of change of the function at x = 2.

Finally, calculate the differential dy using the formula dy = f'(x) * dx. Substitute f'(2) and dx = 0.2 into this formula to find dy.

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Struggling with Calculus?

If f(x)=x3−8x+6f\left(x\right)=x^3-8x+6f(x)=x3−8x+6, find the differential dydydy when x = 2 and dx=0.2dx = 0.2dx=0.2.

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Differentials practice set

If $f\left(x\right)=x^3-8x+6$ , find the differential $dy$ when x = 2 and $dx = 0.2$ .