# How do you differentiate #f(x)= e^x/(e^(x^2) +2x )# using the quotient rule?

[Ans]

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To differentiate the function ( f(x) = \frac{e^x}{e^{x^2} + 2x} ) using the quotient rule, follow these steps:

- Identify the numerator ( u(x) = e^x ) and the denominator ( v(x) = e^{x^2} + 2x ).
- Apply the quotient rule formula: ( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} ).
- Find the derivatives of ( u(x) ) and ( v(x) ):
- ( u'(x) = e^x ) (derivative of ( e^x )).
- ( v'(x) = (2x)e^{x^2} + 2 ) (derivative of ( e^{x^2} + 2x ) using the chain rule).

- Substitute these values into the quotient rule formula:
- ( f'(x) = \frac{(e^x)(e^{x^2} + 2x) - (e^x)((2x)e^{x^2} + 2)}{[e^{x^2} + 2x]^2} ).

- Simplify the expression if necessary.

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