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Given f(x) = e^3x
We need to find the first derivative f'(x) using the chain rule.
We will assume that u= 3x ==> u' = 3
==> f(x) = e^u
==> Now we will differentiate:
==> f(x) = (e^u)' = u' * e^u du
==> Now we will substitute with x.
==> f(x) = 3 ** e^3x = 3*e^3x
Thenthe derivative of f(x) is f'(x) = 3*e^3x
We have to find the chain rule to find the derivative of f(x) = e^3x.
The chain rule for the function f(x) = g(h(x)) gives f'(x) = g'(h(x))*h'(x).
Here f(x) = e^3x
f'(x) = e^3x * (3x)'
=> e^3x * 3
The required derivative is 3*e^3x.
Since the given function is a composed function, we'll apply chain rule to find it's derivative:
We'll have (u(v(x)) = e^3x
u(v) = e^v => u'(v) = e^v (u is differentiated with respect to v)
v(x) = 3x => v'(x) = 3 (v is differentiated with respect to x).
f'(x) = (u(v(x))' = v'(x)*e^v
f'(x) = 3*e^3x
The result of differentiation of the given function is: f'(x) = 3e^3x.
Remember that the derivative of e^u is the derivative of u times e^u
f'(x) = 3e^3x
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