# Calculate the derivative of # y = (x^2+2)^2(x^4+4)^4 # using logarithms?

Taking Natural Logarithms:

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To calculate the derivative of (y = (x^2+2)^2(x^4+4)^4) using logarithms, we'll use the property that the natural logarithm can be applied to simplify expressions involving multiplication and exponentiation.

[ y = (x^2+2)^2(x^4+4)^4 ]

[ \ln(y) = \ln\left((x^2+2)^2(x^4+4)^4\right) ]

[ \ln(y) = \ln((x^2+2)^2) + \ln((x^4+4)^4) ]

[ \ln(y) = 2\ln(x^2+2) + 4\ln(x^4+4) ]

Now, we'll differentiate both sides with respect to (x):

[ \frac{1}{y} \frac{dy}{dx} = 2\frac{1}{x^2+2} \frac{d}{dx}(x^2+2) + 4\frac{1}{x^4+4} \frac{d}{dx}(x^4+4) ]

[ \frac{1}{y} \frac{dy}{dx} = \frac{2x}{x^2+2} + \frac{16x^3}{x^4+4} ]

Finally, multiply both sides by (y) to solve for (\frac{dy}{dx}):

[ \frac{dy}{dx} = y\left(\frac{2x}{x^2+2} + \frac{16x^3}{x^4+4}\right) ]

[ \frac{dy}{dx} = (x^2+2)^2(x^4+4)^4\left(\frac{2x}{x^2+2} + \frac{16x^3}{x^4+4}\right) ]

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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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