Use logarithmic differentiation to find the derivative of the function. y = (x3 + 2)2(x4 + 4)4
Solution:
Given, y = (x3 + 2)2(x4 + 4)4
We have to find the derivative of the function using logarithmic differentiation.
Taking natural log on both sides,
ln y = ln (x3+2)2 (x4+4)4
Using log product property,
ln (a.b) = ln a + ln b
ln y = ln (x3 + 2)2 + ln (x4 + 4)4
Differentiating both sides using chain rule,
y’ = {[6x2 ln(x3 + 2)]/(x3 + 2)} + {[16x3 ln(x4 + 4)4]/(x4 + 4)} × y
Put the value of y in y’
y’ = {[6x2 ln(x3 + 2)]/(x3 + 2)} + {[16x3 ln(x4 + 4)4]/(x4 + 4)} {(x3 + 2)2(x4 + 4)4}
y’ = {[6x2 ln(x3 + 2)]/(x3 + 2)}{(x3 + 2)2(x4 + 4)4} + {[16x3 ln(x4 + 4)4]/(x4 + 4)} {(x3 + 2)2(x4 + 4)4}
y’ = {[6x2 ln(x3 + 2)](x4 + 4)4(x3 + 2)} + {[16x3 ln(x4 + 4)4](x4 + 4)3(x3 + 2)2}
y’ = (x4 + 4)3(x3 + 2)[{6x2 ln(x3 + 2)(x4 + 4)} + {16x3 ln(x4 + 4)4(x3 + 2)}]
Therefore, the derivative is y’ = (x4 + 4)3(x3 + 2)[{6x2 ln(x3 + 2)(x4 + 4)} + {16x3 ln(x4 + 4)4(x3 + 2)}]
Use logarithmic differentiation to find the derivative of the function. y = (x3 + 2)2(x4 + 4)4
Summary:
The derivative of the function y = (x3 + 2)2(x4 + 4)4 using logarithmic differentiation is y’ = (x4 + 4)3(x3 + 2)[{6x2 ln(x3 + 2)(x4 + 4)} + {16x3 ln(x4 + 4)4(x3 + 2)}]
visual curriculum