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How do you differentiate #x^(6x)#?

Answer 1

Samantha Adkison

You can write this as

#x^(6*x)=e^(6x*lnx)#

Hence

#d(x^(6x))/dx=d(e^(6xlnx)/dx)=e^(6xlnx)(6lnx+6)= 6x^(6x)(lnx+1)#

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Answer 2

Paisley Johnson

To differentiate (x^{6x}), we use the chain rule, which states that if (f(x) = g(h(x))), then (f'(x) = g'(h(x)) \cdot h'(x)).

Let (g(u) = u^6) and (h(x) = x).

Differentiate (g(u)) with respect to (u) to find (g'(u)):

[g'(u) = 6u^5]

Now, differentiate (h(x)) with respect to (x) to find (h'(x)):

[h'(x) = 1]

Apply the chain rule:

[f'(x) = g'(h(x)) \cdot h'(x)] [f'(x) = 6(x^6)^5 \cdot 1] [f'(x) = 6x^5 \cdot x^6] [f'(x) = 6x^5 \cdot x^6] [f'(x) = 6x^{5+6}] [f'(x) = 6x^{11}]

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Answer from HIX Tutor

When evaluating a one-sided 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.