Advanced Calculus (I)
WEN-CHINGLIEN
Department of Mathematics National Cheng Kung University
WEN-CHINGLIEN Advanced Calculus (I)
4.2 Differentiability Theorems
Theorem
Let f and g be real function andα∈R.If f and g are differentiable at a, then f+g,αf , f ·g, and (when g(a)6=0) f
g are all differentiable at a. In fact, (f +g)0(a) =f0(a) +g0(a),
(αf)0(a) =αf0(a),
(f ·g)0(a) =g(a)f0(a) +f(a)g0(a).
f
g 0
(a) = g(a)f0(a)−f(a)g0(a) g2(a)
WEN-CHINGLIEN Advanced Calculus (I)
4.2 Differentiability Theorems
Theorem
Let f and g be real function andα∈R.If f and g are differentiable at a, then f+g,αf , f ·g, and (when g(a)6=0) f
g are all differentiable at a. In fact, (f +g)0(a) =f0(a) +g0(a),
(αf)0(a) =αf0(a),
(f ·g)0(a) =g(a)f0(a) +f(a)g0(a).
f
g 0
(a) = g(a)f0(a)−f(a)g0(a) g2(a)
WEN-CHINGLIEN Advanced Calculus (I)
4.2 Differentiability Theorems
Theorem
Let f and g be real function andα∈R.If f and g are differentiable at a, then f+g,αf , f ·g, and (when g(a)6=0) f
g are all differentiable at a. In fact, (f +g)0(a) =f0(a) +g0(a),
(αf)0(a) =αf0(a),
(f ·g)0(a) =g(a)f0(a) +f(a)g0(a).
f
g 0
(a) = g(a)f0(a)−f(a)g0(a) g2(a)
WEN-CHINGLIEN Advanced Calculus (I)
4.2 Differentiability Theorems
Theorem
Let f and g be real function andα∈R.If f and g are differentiable at a, then f+g,αf , f ·g, and (when g(a)6=0) f
g are all differentiable at a. In fact, (f +g)0(a) =f0(a) +g0(a),
(αf)0(a) =αf0(a),
(f ·g)0(a) =g(a)f0(a) +f(a)g0(a).
f
g 0
(a) = g(a)f0(a)−f(a)g0(a) g2(a)
WEN-CHINGLIEN Advanced Calculus (I)
4.2 Differentiability Theorems
Theorem
Let f and g be real function andα∈R.If f and g are differentiable at a, then f+g,αf , f ·g, and (when g(a)6=0) f
g are all differentiable at a. In fact, (f +g)0(a) =f0(a) +g0(a),
(αf)0(a) =αf0(a),
(f ·g)0(a) =g(a)f0(a) +f(a)g0(a).
f
g 0
(a) = g(a)f0(a)−f(a)g0(a) g2(a)
WEN-CHINGLIEN Advanced Calculus (I)
4.2 Differentiability Theorems
Theorem
Let f and g be real function andα∈R.If f and g are differentiable at a, then f+g,αf , f ·g, and (when g(a)6=0) f
g are all differentiable at a. In fact, (f +g)0(a) =f0(a) +g0(a),
(αf)0(a) =αf0(a),
(f ·g)0(a) =g(a)f0(a) +f(a)g0(a).
f
g 0
(a) = g(a)f0(a)−f(a)g0(a) g2(a)
WEN-CHINGLIEN Advanced Calculus (I)
Theorem (Chain Rule)
Let f and g be real functions. If f is differentiable at a and g is differentiable at f(a), then g◦f is differentiable at a with
(g◦f)0(a) =g0(f(a))f0(a).
WEN-CHINGLIEN Advanced Calculus (I)
Theorem (Chain Rule)
Let f and g be real functions. If f is differentiable at a and g is differentiable at f(a), then g◦f is differentiable at a with
(g◦f)0(a) =g0(f(a))f0(a).
WEN-CHINGLIEN Advanced Calculus (I)
Thank you.
WEN-CHINGLIEN Advanced Calculus (I)
