Use the technique of logarithmic differentiation to derive the following formula

Question

Use the technique of logarithmic differentiation to derive the following formulas (assuming there are no domain or definition issues).  Show your work and the use of the process needed to solve for full points.

#1   (d/dx)(fgh) = (f')(g)(h)+(f)(g')(h)+(f)(g)(h')

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Use the technique of logarithmic differentiation to derive the following formulas (assuming there are no domain or definition issues). Show your work and the use of the process needed to solve for full points. (d/dx)(fgh) = (f')(g)(h)+(f)(g')(h)+(f)(g)(h')

Explanation / Answer

Let, y = f * g * h

Taking logarithm on both sides:

log y = log ( f * g * h )

log y = log f + log g + log h

Differentiating both sides w.r.t x:

1/y * dy/dx = (1/f * f') + (1/g * g') + (1/h * h')

dy/dx = y * ( (1/f * f') + (1/g * g') + (1/h * h') )

dy/dx = fgh * ( (1/f * f') + (1/g * g') + (1/h * h') )

Thus, dy/dx = (f')(g)(h)+(f)(g')(h)+(f)(g)(h')

Hence, proved

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