当y = x ^ 2x(x-1)^ 3 /(3 + 5x)^ 4时如何找到dy / dx

您如何找到函数[math] \ frac {f(x)} {g(x)} [/ math]的导数?

令[math] y \,= \,[/ math] [math] f(x)^ {g(x)} [/ math]

取双方的自然对数,我们得到

[数学] \ log y \,= \,g(x)\ log f(x)[/数学]

区分双方,我们得到,

[数学] \ frac {1} {y} \,\ frac {dy} {dx} \,= \,\ frac {1} {f(x)} \,g(x)\,\ frac {d} {dx} \ left(f(x)\ right)\,+ \,\ log f(x)\,\ frac {d} {dx} \ left(g(x)\ right)[/数学]

[math] \ implies \ qquad \ frac {dy} {dx} \,= \,y \ left [\ frac {1} {f(x)} \,g(x)\,\ frac {d} {dx } \ left(f(x)\ right)\,+ \,\ log f(x)\,\ frac {d} {dx} \ left(g(x)\ right)\ right] [/ math]

[math] \ implies \ qquad \ frac {dy} {dx} \,= \,f(x)^ {g(x)} \ left [\ frac {1} {f(x)} \,g(x )\,\ frac {d} {dx} \ left(f(x)\ right)\,+ \,\ log f(x)\,\ frac {d} {dx} \ left(g(x)\对)\ right] [/数学]

找出f(x)和g(x)的导数并解决问题。