what derivate rule this two function is using

Question

y=5+e^4t+5^2
y=-2x^-1 -1/x^3 +3x^-4

Answer 1

The first function, y=5+e^4t+5^2, is using the derivative rule for constant terms and the derivative rule for exponential functions.

The derivative of a constant term (5^2) is 0.

The derivative of e^4t can be found using the chain rule:

d(e^4t)/dt = 4*e^4t

So, the derivative of y with respect to t is:

dy/dt = 0 + 4*e^4t + 0 = 4*e^4t

Therefore, the derivative rule used for this function is the derivative rule for constant terms and the derivative rule for exponential functions.

For the second function, y=-2x^-1 -1/x^3 +3x^-4, it is using the derivative rule for negative exponents and the derivative rule for power functions.

The derivative of x^-1 can be found using the power rule:

d(x^-1)/dx = -1*x^(-1-1) = -x^-2 = -1/x^2

The derivative of x^-3 can also be found using the power rule:

d(x^-3)/dx = -3*x^(-3-1) = -3*x^-4 = -3/x^4

Similarly, the derivative of x^-4 can be found:

d(x^-4)/dx = -4*x^(-4-1) = -4*x^-5 = -4/x^5

Therefore, the derivative of y with respect to x is:

dy/dx = -2*(-1/x^2) -1(-3/x^4) +3(-4/x^5) = 2/x^2 + 3/x^4 - 12/x^5

So, the derivative rule used for this function is the derivative rule for negative exponents and the derivative rule for power functions.