Answer :

Final answer:

The derivatives of the given functions are Y' = 86x for Y = 43x^2 + 71, F'(X) = -2X for X^2 <= 1 and F'(X) = 2X for X^2 > 1 for F(X) = |1-X^2|, and F'(T) = t*cos(t) for F(T) = Tsint + cost.

Explanation:

To find the derivatives of the provided functions, we use the rules of differentiation. The rules of differentiation are arithmetic tools that provide equations to determine the derivative, or rate of change, of a function.

(A) Y = 43x2 + 71

Using the power rule of differentiation, which states that the derivative of x^n is n*x^(n-1), the derivative is:

Y' = 2*43*x(2-1) + 0 = 86x

(B) F(X) = |1-X2|

For absolute value functions, we separate it into two cases: one for ((1-X2) >= 0) and another for ((1-X2) < 0).

So, the derivative will be:

F'(X) = -2X for X2 <= 1 and F'(X) = 2X for X2 > 1

(C) F(T) = Tsint + cost

Here, we use the product rule (uv' + vu') and chain rule. Hence the derivative is:

F'(T) = (1*sin(t) + t*cos(t)) - sin(t) = t*cos(t)

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