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.

5. Applications of Derivatives

Differentials

Business Calculus

5. 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

Step 1: Recall the formula for the differential. The differential is given by $ dy = f'(x) \cdot dx $, where $ f'(x) $ is the derivative of the function $ f(x) $ with respect to $ x $, and $ dx $ is the small change in $ x $.

Step 2: Compute the derivative of $ f(x) = x^3 - 8x + 6 $. Using the power rule, the derivative is $ f'(x) = 3x^2 - 8 $.

Step 3: Evaluate $ f'(x) $ at $ x = 2 $. Substitute $ x = 2 $ into $ f'(x) = 3x^2 - 8 $ to find the slope of the tangent line at that point.

Step 4: Multiply the value of $ f'(2) $ by $ dx = 0.2 $ to compute $ dy $. Use the formula $ dy = f'(x) \cdot dx $.

Step 5: The result of $ dy $ represents the approximate change in $ y $ when $ x $ changes by $ dx = 0.2 $. Compare this value to the provided answer choices to identify the correct one.