What are the first and second derivatives of #f(x)=5^((x^5)9x)#?
Following the rule to differentiate exponential functions:
As for the second derivative, we must see we have a three terms product. To differentiate it, we'll consider two of them as one, in a sort of chain rule, and then derivate these two as well, as shown in the formula below:
As zero cancels the product of the second term, we have left:
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To find the first and second derivatives of ( f(x) = 5^{(x^5  9x)} ):

First derivative: [ f'(x) = (\ln 5) \cdot 5^{(x^5  9x)} \cdot (5x^4  9) ]

Second derivative: [ f''(x) = (\ln 5)^2 \cdot 5^{(x^5  9x)} \cdot ((5x^4  9)^2 + 20x^3) ]
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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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