(Calculus)Differentiation and its application

[Jojeti]

Hello! I was wondering if anyone could assist me on a few calculus questions, any help would be appreciated, thanks

Determine the differentials of the following functions and calculate the gradient at the given point:
a) y= x2-2+ 1/x2 (x= -2)
b) y= 2ex+3x2+2 (x= 0)
c) y= sin2x + tanx (x= 45̊)

[MAS1]

Originally Posted by Jojeti

Hello! I was wondering if anyone could assist me on a few calculus questions, any help would be appreciated, thanks

Determine the differentials of the following functions and calculate the gradient at the given point:
a) y= x2-2+ 1/x2 (x= -2)
b) y= 2ex+3x2+2 (x= 0)
c) y= sin2x + tanx (x= 45̊)

I am not completely positive about the notations you have used. Here is what I think they are.

a. y = x^2 - 2 + (1/x^2) (x = -2)
y = x^2 - 2 + x ^(-2)
dy/dx = 2x - 2x^(-3)
dy/dx (at x = -2) = 2(-2) - 2((-2)^(-3)) = -4 - (-1/4) = -3 3/4 = -15/4

b. y = 2e^x + 3x^2 + 2 (x = 0)
dy/dx = 2e^x + 6x
dy/dx (at x = 0) = 2(e^0) + 6(0) = 2 + 0 = 2

c. y = sin(2x) + tan(x) (x = 45 degs.)
dy/dx = 2cos(2x) + (sec(x))^2
dy/dx (at x = 45) = 2cos(90) + (sec(45)(sec(45)) = 0 + 2 = 2

I hope that is how you meant your notation for the different functions.