91Ó°ÊÓ

Three functions $f$its derivatives are just constant multiples of$f$.

Need to write three functions its derivatives are just constant multiples of$f$.

All exponential functions have the property that their derivatives are constant multiplies of the original function.

Example: role="math" localid="1663321382981" $e^{2x},e^{3x},e^{4x}$

Therefore, the three functions its derivatives are just constant multiples of$f$is$e^{2x},e^{3x},e^{4x}$.

Three functions its derivatives are algebraic but the function are transcendental.

Need to write three functions its derivatives are algebraic but the function are transcendental.

Logarithmic functions are transcendental but their derivatives are algebraic.

Example: $\frac{d}{dx}\left({\log }_{b}x\right)=\frac{1}{\ln bx},\frac{d}{dx}(\ln x)=\frac{1}{x},\frac{d}{dx}\left(\ln \left|x\right|\right)=\frac{1}{x}$.

Thus, the three functions its derivatives are algebraic, but the function are transcendental is$\frac{d}{dx}\left({\log }_{b}x\right)=\frac{1}{\ln bx},\frac{d}{dx}(\ln x)=\frac{1}{x},\frac{d}{dx}\left(\ln \left|x\right|\right)=\frac{1}{x}$.

Three functions it’s would be difficult or impossible to find without the method of logarithmic differentiation.

Need to write three functions it’s would be difficult or impossible to find without the method of logarithmic differentiation.

Take the derivatives of a function that involves the variables in both the base and exponent must use the logarithmic differentiation.

Example: $x^{\ln x},{(2x+1)}^{3x},{(\ln x)}^{\ln x}$

Therefore, the three functions it’s would be difficult or impossible to find without the method of logarithmic differentiation is$x^{\ln x},{(2x+1)}^{3x},{(\ln x)}^{\ln x}$.