find the derivative
f(x)= e^1/x^2�ãx
To find the derivative of the function f(x) = e^(1/x^2) * x, we can use the product rule and the chain rule.
Let's break down the steps:
Step 1: Apply the Product Rule
The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)
In our case, u(x) = e^(1/x^2) and v(x) = x.
So, applying the product rule, we get:
f'(x) = u'(x) * v(x) + u(x) * v'(x)
Step 2: Differentiate u(x)
To differentiate u(x) = e^(1/x^2), we can use the chain rule. The chain rule states that if we have a composite function g(f(x)), then the derivative is given by:
(d/dx)(g(f(x))) = g'(f(x)) * f'(x)
Let's differentiate u(x) = e^(1/x^2):
Let f(x) = 1/x^2.
g(f) = e^f.
Applying the chain rule, we have:
u'(x) = g'(f(x)) * f'(x)
To find g'(f(x)), we can differentiate g(f) = e^f as a simple exponential function:
g'(f) = e^f
Since f(x) = 1/x^2, we have:
g'(f(x)) = e^(1/x^2)
Step 3: Differentiate v(x)
v(x) = x is a simple polynomial function, and its derivative is straightforward to calculate:
v'(x) = 1
Step 4: Substitute the derivatives into the product rule equation
Now that we have u'(x) = e^(1/x^2) and v'(x) = 1, we can substitute them into the product rule equation:
f'(x) = u'(x) * v(x) + u(x) * v'(x)
f'(x) = e^(1/x^2) * x + e^(1/x^2) * 1
Simplifying further, we get:
f'(x) = e^(1/x^2) * (x + 1)