Differentitate y = e^x^2 -^5^x +^4
so far i have
dy/dx = 2x e^(x^2) - 5^(x)* ln 5 +
can you help with the rest?
Nope.
y=e^u -5^z I don't understand the rest.
y'=ue^u u' - ln5 * z'
where u' would be 2x and z' would be 1
See
http://www.jiskha.com/display.cgi?id=1245868785
To differentiate the given function y = e^x^2 - 5^x + 4, you have correctly calculated the derivative up to this point:
dy/dx = 2x e^(x^2) - 5^(x) ln(5) +
Now, let's complete the differentiation of the function:
To find the derivative of e^x^2, we need to use the chain rule since the exponent contains x^2. The chain rule states that if you have a function g(x) embedded within another function f(g(x)), then the derivative of f(g(x)) is given by f'(g(x)) * g'(x).
In this case, g(x) = x^2, and f(u) = e^u, where u = x^2. Hence, f(g(x)) = e^(x^2).
Taking the derivative of e^(x^2) with respect to x requires applying the chain rule:
d/dx(e^(x^2)) = d/du(e^u) * d/dx(x^2)
= e^u * 2x
= 2xe^(x^2)
Therefore, the first term of the derivative is correctly determined as:
2x e^(x^2)
Now, let's move on to the second term, -5^x. To differentiate this term, we consider it as a power of a constant -5:
d/dx(-5^x) = ln(-5) * d/dx(e^(x ln(-5)))
= ln(-5) * e^(x ln(-5)) * ln(-5)
Note that ln(-5) is the natural logarithm of -5 and is a complex value. The derivative of -5^x with respect to x is given by:
ln(-5) * e^(x ln(-5)) * ln(-5)
Finally, the complete derivative of the given function is:
dy/dx = 2x e^(x^2) - 5^(x) ln(5) + ln(-5) * e^(x ln(-5)) * ln(-5)