问题1:从第一原理中得出以下函数的派生:
(i)-x
解决方案:
f(x) = -x
f(x+h) = -(x+h)
From the first principle,
f'(x) = -1
(ii)(-x) -1
解决方案:
f(x) = (-x)-1 =
f(x+h) = (-(x+h))-1 =
From the first principle,
(iii)罪(x + 1)
解决方案:
f(x) = sin(x+1)
f(x+h) = sin((x+h)+1)
From the first principle,
Using the trigonometric identity,
sin A – sin B = 2 cos sin
Multiply and divide by 2, we have
f'(x) = cos (x+1) (1)
f'(x) = cos (x+1)
(iv)
解决方案:
Here,
From the first principle,
Using the trigonometric identity,
cos a – cos b = -2 sin sin
Multiplying and diving by 2,
找到以下函数的导数(可以理解,a,b,c,d,p,q,r和s是固定的非零常数,m和n是整数):
问题2:(x + a)
解决方案:
f(x) = x+a
Taking derivative both sides,
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题3:
解决方案:
Taking derivative both sides,
Using the product rule, we have
(uv)’ = uv’+u’v
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题4:(ax + b)(cx + d) 2
解决方案:
f(x) = (ax+b) (cx+d)2
Taking derivative both sides,
Using the product rule, we have
(uv)’ = uv’+u’v
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题5:
解决方案:
Taking derivative both sides,
Using the quotient rule, we have
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题6:
解决方案:
Taking derivative both sides,
Using the quotient rule, we have
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题7:
解决方案:
Taking derivative both sides,
Using the quotient rule, we have
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题8:
解决方案:
Taking derivative both sides,
Using the quotient rule, we have
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题9:
解决方案:
Taking derivative both sides,
Using the quotient rule, we have
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题10:
解决方案:
Taking derivative both sides,
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题11:
解决方案:
Taking derivative both sides,
As, the derivative of xn is nxn-1 and derivative of constant is 0.
问题12:(ax + b) n
解决方案:
f(x) = (ax+b)n
f(x+h) = (a(x+h)+b)n
f(x+h) = (ax+ah+b)n
From the first principle,
Using the binomial expansion, we have
问题13:(ax + b) n (cx + d) m
解决方案:
f(x) = (ax+b)n (cx+d)m
Taking derivative both sides,
Using the product rule, we have
(uv)’ = uv’+u’v
Let’s take, g(x) = (cx+d)m
g(x+h) = (c(x+h)+d)m
g(x+h) = (cx+ch+d)m
From the first principle,
Using the binomial expansion, we have
So, as
问题14:罪(x + a)
解决方案:
f(x) = sin(x+a)
f(x+h) = sin((x+h)+a)
From the first principle,
Using the trigonometric identity,
sin A – sin B = 2 cos sin
Multiply and divide by 2, we have
问题15:cosec x cot x
解决方案:
f(x) = cosec x cot x
Taking derivative both sides,
Using the product rule, we have
(uv)’ = uv’+u’v
f'(x) = cot x (-cot x cosec x) + (cosec x) (-cosec2 x)
f'(x) = – cot2 x cosec x – cosec3 x