How do you find the derivative of #x^(2x)#?

Answer 1
The answer is: #y'=2e^(2xlnx)(lnx+1)#.

Instead of remembering a complicate formula, we use these logarithmic properties:

#a=e^lna# or, better: #a^b=e^ln(a^b)=e^(blna)#.

So our function becomes:

#y=e^(2xlnx)#

and

#y'=e^(2xlnx)(2*1*lnx+2x*1/x)=2e^(2xlnx)(lnx+1)#.
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Answer 2

To find the derivative of (x^{2x}), use the exponential rule for differentiation, which states that ( \frac{d}{dx}(a^x) = a^x \ln(a) ), where (a) is a constant. Apply this rule by treating (x^{2x}) as (e^{2x \ln(x)}) and then differentiate using the chain rule.

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Answer 3

To find the derivative of ( x^{2x} ), you can use the logarithmic differentiation method.

  1. Take the natural logarithm (ln) of both sides of the expression: [ \ln(y) = \ln(x^{2x}) ]

  2. Apply logarithmic properties to simplify: [ \ln(y) = 2x \ln(x) ]

  3. Differentiate both sides implicitly with respect to ( x ): [ \frac{1}{y} \cdot \frac{dy}{dx} = 2 \ln(x) + 2x \cdot \frac{1}{x} ] [ \frac{dy}{dx} = y \left(2 \ln(x) + 2 \right) ]

  4. Since ( y = x^{2x} ), substitute this back into the equation: [ \frac{dy}{dx} = x^{2x} \left(2 \ln(x) + 2 \right) ]

Thus, the derivative of ( x^{2x} ) with respect to ( x ) is ( x^{2x} (2 \ln(x) + 2) ).

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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.

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.

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.

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.

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