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Find dy/dx using logarithmic differentiation.
y=(x^3)sqrt(1+(x^2))
To find dy/dx using logarithmic differentiation, follow these steps:
1. Take the natural logarithm of both sides of the equation. This step helps us simplify the expression by using logarithmic properties.
ln(y) = ln((x^3)sqrt(1+(x^2)))
2. Apply the logarithmic rules to simplify the expression further.
ln(y) = ln(x^3) + ln(sqrt(1+(x^2)))
ln(y) = 3ln(x) + ln(1+(x^2))/2
3. Differentiate both sides of the equation with respect to x.
Using the chain rule, differentiate ln(y) with respect to x:
(d/dx) ln(y) = (d/dx) (3ln(x) + ln(1+(x^2))/2)
Using the chain rule, the derivative of ln(y) with respect to y is (1/y) * (dy/dx), and the derivative of ln(x) with respect to x is (1/x) * (dx/dx) = (1/x). Therefore, the left side becomes:
(1/y) * (dy/dx) = (3/x) + (1/2) * (2x/(1+(x^2)))
4. Solve for dy/dx.
(1/y) * (dy/dx) = (3/x) + (x/(1+(x^2)))
Multiply both sides by y:
dy/dx = y * [(3/x) + (x/(1+(x^2)))]
Substitute y = (x^3)sqrt(1+(x^2)) into the equation:
dy/dx = (x^3)sqrt(1+(x^2)) * [(3/x) + (x/(1+(x^2)))]
Therefore, dy/dx = (x^3)sqrt(1+(x^2)) * [(3/x) + (x/(1+(x^2)))]