Question

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
tell me hwat rule is used and the working out

Answer 1

y=5+e^(4t)+5^2
dy/dt = 4e^(4t)
But I suspect a typo, seeing that pesky 5^2 at the end.

y=-2x^-1 -1/x^3 +3x^-4
dy/dx = 2/x^2 + 3/x^4 - 20/x^5
there is no chain rule involved!

Answer 2

The first function, y=5+e^4t+5^2, does not involve any derivative rules, as it does not contain any independent variable being differentiated.

The second function, y=-2x^-1 -1/x^3 +3x^-4, involves several derivative rules:

1. Quotient rule: The term -2x^-1 can be rewritten as -2/x, and -1/x^3 can be rewritten as -x^-3. Both these terms involve the power rule, as well as the constant multiple rule.

2. Chain rule: The term 3x^-4 involves the power rule, and since the power of x is negative, the chain rule is applied to differentiate the term properly.

Answer 3

Apologies for the confusion in the previous response. You are correct.

For the function y=5+e^(4t)+5^2, the derivative is dy/dt = 4e^(4t). The derivative is obtained using the chain rule, where the derivative of e^(4t) is 4e^(4t).

For the function y=-2x^-1 -1/x^3 +3x^-4, the derivative is dy/dx = 2/x^2 + 3/x^4 - 20/x^5. The terms -2x^-1, -1/x^3, and 3x^-4 are differentiated using the power rule, and the derivative of each term is obtained as 2/x^2, 3/x^4, and -20/x^5, respectively. There is no chain rule involved in this case.