How do you find the derivative of #x^(2x)#?
Instead of remembering a complicate formula, we use these logarithmic properties:
So our function becomes:
and
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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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To find the derivative of ( x^{2x} ), you can use the logarithmic differentiation method.

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

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

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) ]

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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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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