what derivate rule this two function is using
y=5+e^4t+5^2
y=-2x^-1 -1/x^3 +3x^-4
example quotient, product and chain
The first function: y = 5 + e^(4t) + 5^2
The derivative of the sum of functions uses the rule:
d/dx (f(x) + g(x)) = f'(x) + g'(x)
Therefore, applying this rule to the first function:
dy/dt = d/dt (5) + d/dt (e^(4t)) + d/dt (5^2)
= 0 + 4e^(4t) + 0
= 4e^(4t)
The second function: y = -2x^(-1) - 1/x^3 + 3x^(-4)
The derivative of the difference of functions uses the rule:
d/dx (f(x) - g(x)) = f'(x) - g'(x)
Therefore, applying this rule to the second function:
dy/dx = d/dx (-2x^(-1)) - d/dx (1/x^3) + d/dx (3x^(-4))
= -(-1)x^(-2) - (-3)x^(-4-1) + (3)(-4)x^(-4-1)
= 2x^(-2) + 3x^(-4+1) - 12x^(-5)
= 2/x^2 + 3/x - 12/x^5