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Y=(x3x+5​)2x Use Logarithmic differentiation to find dy/dx.

User Ding Peng
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Explanation:

To find the derivative of the function y = (x^3 * e^(x+5))^2x using logarithmic differentiation, we follow these steps:

Take the natural logarithm (ln) of both sides of the equation:

ln(y) = ln((x^3 * e^(x+5))^2x)

Apply the properties of logarithms to simplify the expression:

ln(y) = 2x * ln(x^3 * e^(x+5))

Expand the logarithm using the properties of logarithms:

ln(y) = 2x * (ln(x^3) + ln(e^(x+5)))

Simplify the logarithms:

ln(y) = 2x * (3ln(x) + (x+5))

Differentiate both sides of the equation with respect to x:

(d/dx) ln(y) = (d/dx) [2x * (3ln(x) + (x+5))]

Apply the chain rule and simplify:

(dy/y) = [2 * (3ln(x) + (x+5))] + [2x * (3/x) + 1]

Rearrange the equation to solve for dy/dx:

dy/dx = y * [2 * (3ln(x) + (x+5))] + [2x * (3/x) + 1]

Finally, we can substitute back y = (x^3 * e^(x+5))^2x into the equation to obtain the derivative in terms of x.

User Gextra
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