1. is the derivative of the function f(x)=5e^(4x-9z) f'(x)= -25e^(4x-9z)?
2. Is the derivative of the function f(x)=e^(2x)-e^(-2x) f'(x)=2e^(2x)+2e^(-2x)?
3. If f(x)=e^-x^-1, would f'(2)=-1/4e^(1/4)?
1) if you have a function f(x) , what is that 9z doing there in the exponent?
2) yes
3.) is that a stair-case exponent or is it
f(x) = (e^x)^-1 ?
= e^-x ?
on the other hand, Wolfram read your equation as
y = e^(1/x)
http://www.wolframalpha.com/input/?i=plot+y+%3D+e%5Ex%5E-1
in that case,
dy/dx = (-1/x^2) e^(1/x)
when x = 2
dy/dx = (-1/4)e^(1/2)
or -(1/4) √e
1.)sorry i meant to put x not z. so 9x.
3.)yes it is a stair case exponent
1) in that case why not just write the exponent as -5x ?
y = 5 e^(-5x)
y' = -25 e^(-5x)
3. but your answer resembles the other interpretation, for which I gave you a solution.
I stand by that answer.
To find the derivative of a function, you can use the rules of differentiation. The two main rules to keep in mind are the power rule and the chain rule.
1. For the function f(x) = 5e^(4x-9z), we can use the chain rule. The chain rule states that if you have a composition of functions, such as f(g(x)), the derivative is equal to the derivative of the outside function (f'(g(x))) times the derivative of the inside function (g'(x)). In this case, the outside function is e^(4x-9z) and the inside function is 4x-9z.
To find f'(x), we can first find the derivative of e^(4x-9z), which is e^(4x-9z) multiplied by the derivative of the exponent, which is 4. So, f'(x) = 5e^(4x-9z) * 4 = 20e^(4x-9z).
Therefore, the correct derivative is f'(x) = 20e^(4x-9z), not -25e^(4x-9z).
2. For the function f(x) = e^(2x) - e^(-2x), we can again use the chain rule. The first term of the function is e^(2x), and the second term is e^(-2x).
To find f'(x), we can find the derivative of both terms separately. The derivative of e^(2x) is e^(2x) multiplied by the derivative of the exponent, which is 2. Similarly, the derivative of e^(-2x) is e^(-2x) multiplied by the derivative of the exponent, which is -2.
So, f'(x) = e^(2x) * 2 - e^(-2x) * 2 = 2e^(2x) - 2e^(-2x).
Therefore, the correct derivative is f'(x) = 2e^(2x) - 2e^(-2x), not 2e^(2x) + 2e^(-2x).
3. For the function f(x) = e^(-x^(-1)), we need to use the chain rule and the power rule. The chain rule tells us that the derivative of e^(-x^(-1)) is e^(-x^(-1)) multiplied by the derivative of the inside function, which is -1/x^2. The power rule tells us that the derivative of x^(-1) is -1/x^2.
To find f'(2), we substitute x = 2 into the derivative expression we obtained. So, f'(2) = e^(-2^(-1)) * (-1/(2^2)) = e^(-1/2) * (-1/4) = -1/4e^(1/2).
Therefore, the correct value of f'(2) is -1/4e^(1/2), not -1/4e^(1/4).