高数中,复合函数的求导方式是怎么推导出来的?

Question

高数中,复合函数的求导方式是怎么推导出来的?

Answer 1

定义
f'(g(x))
f(g(x+h))-f(g(x))
=lim --------------------------
h->0 h
f(g(x+h))-f(g(x)) g(x+h)-g(x)
=lim -------------------------- * -----------------------
h->0 g(x+h)-g(x) h
f(g(x+h))-f(g(x)) g(x+h)-g(x)
=lim -------------------------- * lim -----------------------
h->0 g(x+h)-g(x) h->0 h
第一个极限令y'=g(x+h),y=g(x),显然y'->y当h->0
f(y')-f(y) g(x+h)-g(x)
=lim --------------- * lim -----------------------
y'->y y'-y h->0 h
=f'(y)*g'(x)
=f'(g(x))*g'(x)