How do you differentiate #f(x)=x(1+e^(x^2))^(1/5)# using the chain rule?
First, use the product rule:
Find each derivative separately.
The following requires heavy use of the chain rule:
Plug this back in.
Multiply the first term for a common denominator.
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To differentiate ( f(x) = x(1 + e^{x^2})^{1/5} ) using the chain rule, we proceed as follows:

Identify the outer function and the inner function. Outer function: ( u = u(v) = v^{1/5} ) Inner function: ( v = v(x) = 1 + e^{x^2} )

Differentiate the outer function with respect to its variable. ( \frac{du}{dv} = \frac{1}{5}v^{4/5} )

Differentiate the inner function with respect to its variable. ( \frac{dv}{dx} = \frac{d}{dx}(1 + e^{x^2}) = 0 + e^{x^2} \cdot \frac{d}{dx}(x^2) = 2xe^{x^2} )

Apply the chain rule: Multiply the derivatives of the outer and inner functions. ( \frac{df}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx} = \frac{1}{5}(1 + e^{x^2})^{4/5} \cdot 2xe^{x^2} )

Simplify the expression if necessary.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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