Derive the Equation

Question

Let y=x^7-4x^5+8x^3-2x-7

Then y(2) is ---

Then (dy)/(dx) is --- 

(d(2))/(dx) is --- 

The second derivative of y is --- 

and y''(2) is -----

Answer 1

y = x⁷ - 4x⁵ + 8x³ - 2x - 7

(1) To find y(2), we can plug in 2 for x:

y(2) = (2)⁷ - 4(2)⁵ + 8(2)³ - 2(2) - 7

= 128 - 4(32) + 8(8) - 4 - 7

= 128 - 128 + 64 - 4 - 7

= 53

y(2) = 53

(2) To find the derivative dy/dx, we can differentiate using the power rule which states:

d/dx(xⁿ) = nx^n-1

To apply the power rule, you basically bring the exponent to the front of the variable and multiply by its coefficient and subtract one from the power.

Lets apply it:

y = x⁷ - 4x⁵ + 8x³ - 2x - 7

dy/dx = 7x⁶ - (5)(4)x⁴ + (3)(8)x² - 2

dy/dx = 7x⁶ - 20x⁴ + 24x² - 2

(3) d(2)/dx means derivative of a constant and the derivative of any constant is 0. So:

d(2)/dx = 0

(4) In order to find the second derivative of y, we can take its first derivative which we found in step 2 and differentiate it again:

First derivative is:

y' = 7x⁶ - 20x⁴ + 24x² - 2

Lets differentiate it again using the power rule:

y'' = (6)(7)x⁵ - (4)(20)x³ + (2)(24)x

y'' = 42x⁵ - 80x³ + 48x

To now find y''(2), we can simply take the second derivative that we just found and plug in 2 for x:

y''(2) = 42(2)⁵ - 80(2)³ + 48(2)

= 42(32) - 80(8) + 96

= 1344 - 640 + 96

y''(2) = 800

Answer 2

y(20 = 53

 

y' = 7x⁶ - 20x⁴ + 16x² - 2

 

y'' = 42x⁵ - 80x³ + 32x