derivatives of trigonometric functions - sinx, cosx, tanx, cotx, secx, cosecx and so on.
sin(x), cos(x) and tan(x) are basic trigonometric functions. cot(x), sec(x) and cosec(x) can derived from three basic trigonometric functions.
tan(x)=sin(x)cos(x)
cot(x)=1tan(x)=1sin(x)cos(x)=cos(x)sin(x)
sec(x)=1cos(x) and
cosec(x)=1sin(x)
This post provides you very useful ideas to find out the derivatives of these trigonometric functions by using the first principle of limit or limit theorem.
To learn basic concept of derivatives, go to the post.
Derivative of function f(x) w.r.t x by using limit theorem:
d(x)dx=limฮxโ0f(x+ฮx)-f(x)ฮx
1. Derivative of sin(x):
Let, y=f(x)=sin(x).....(1)
Also, let ฮx and ฮy be small increments (increase or decrease) in x and y respectively. Then
y+ฮy=sin(x+ฮ)......(2)
Subtract (1) from (2),
ฮy=sin(x+ฮx)-sin(x)
Dividing both sides by ฮx,
ฮyฮx=sin(x+ฮx)-sin(x)ฮx
Now derivative of y w.r.t x as ฮxโ0 is,
dydx=limฮxโ0ฮyฮx=limฮxโ0sin(x+ฮx)-sin(x)ฮx
Using the realtion,
sinA-sinB=2sin(A-B2)cos(A+B2) you get,
sin(x+ฮx)-sin(x)=2sin(ฮx2)cos(2x+ฮx2), so now
dydx=limฮxโ02sin(ฮx2)cos(2x+ฮx2)ฮx
or, dydx=2limฮxโ0sin(ฮx2)ฮx2โ 2cos(2x+ฮx2)
But, limฮxโ0sin(ฮx)ฮx=1 so,
dydx=2limฮx2โ0sin(ฮx2)ฮx2โ 2cos(2x+ฮx2)
or, dydx=2โ 12โ 1โ cos(2x+02)
or, dydx=cos(2x2)=cos(x)
โด dsin(x)dx=cos(x).//
similary you can get, dsin(ax)dx=acos(ax).
2. Derivative of cos(x):
Let, y = f(x) = cos(x) ...(1)
Also, y+ฮy=cos(x+ฮx).....(2)
Subtract (1) from (2),
ฮy=cos(x+ฮx)-cos(x)
or, ฮyฮx=cos(x+ฮx)-cos(x)ฮx
Here using, cos(A)-cos(B)=2sin(A+B2)sin(B-A2) ,
cos(x+ฮx)-cos(x)=2sin(2x+ฮx2)sin(-ฮx2) so,
โด dydx=limฮxโ02sin(2x+ฮx2)sin(-ฮx2)ฮx
or, dydx=2limฮxโ0-sin(ฮx2)2ฮx2sin(2x+ฮx2)
Muliply the value in limit by 12 then,
dydx=2limฮx2โ0-sin(ฮx2)2ฮx2sin(2x+ฮx2)
or, dydx=2-1.12sin(2x+02)
or, dydx=-sin(x)
โด dcos(x)dx=-sin(x)
Similarly you can prove,
dcos(ax)dx=-asin(ax) //
3. Derivative of tan(x):
Let y=f(x)=tan(x)......(1)
Also, y+ฮy=tan(x+ฮx)........(2)
Subtract (1) from (2),
ฮy=tan(x+ฮx)-tan(x)
or, ฮy=sin(x+ฮx)cos(x+ฮx)-sin(x)cos(x)
or, ฮy=sin(x+ฮx)cos(x)-sin(x)cos(x+ฮx)cos(x+ฮx)cos(x)
Using, sin(A - B) = sinA cosB - sinB cosA, so
ฮy=sin(x+ฮx-x)cos(x+ฮx)cos(x)
or, ฮy=sin(ฮx)cos(x+ฮx)cos(x)
Divide by ฮx,
ฮyฮx=sin(ฮx)ฮxโ cos(x+ฮx)cos(x)
Now taking liit as ฮxโ0 on both sides,
dydx=limฮxโ0ฮyฮx=limฮxโ0sin(ฮx)ฮxโ limฮxโ01cos(x+ฮx)cos(x)
or, dydx=1.1cos(x)cos(x)
or, dydx=1cos2(x)=sec2(x)
โด dtan(x)dx=sec2(x)
Similarly, ddxtan(ax)=ddxtan(ax+b)=asec2(x)
Where a and b are constants.//
4. Derivative of cot(x):
Let, y = f(x) = cot(x) ........(1)
Following the same steps as in case of tan(x) using the relation
cos(A-B) = cosA cosB + sinA sinB, you get
ddxcot(x)=-cosec2(x)
Similarly, ddxcot(ax)=ddxcot(ax+b)=-acosec2(x)//
5. Derivative of sec(x):
Let, y=f(x)=sec(x)=1cos(x)......(1)
Also, y+ฮy=1cos(x+ฮx).....(2)
Subtract (1) from (2),
ฮy=1cos(x+ฮx)-1cos(x)
or, ฮy=cos(x)-cos(x+ฮx)cos(x)cos(x+ฮx)
Using the relation, cosA-cosB=2sin(A+B2)sin(B-A2) you have,
cos(x)-cos(x+ฮx)=2sin(2x+ฮx2)sin(ฮx22) so
ฮy=2sin(2x+ฮx2)sin(ฮx2)cos(x+ฮx)cos(x)
Now divide by ฮx and take limit on both sides as ฮx2โ0,
dydx=limฮx2โ02sin(2x+ฮx2)sin(ฮx2)cos(x+ฮx)cos(x)2ฮx2
Using limฮxโ0sin(ฮx)ฮx=1,
โด dydx=limฮx2โ0ฮyฮx=2sin(2x2)2cos(x+0)cos(x)
or, dydx=sin(x)cos(x)cos(x)=sec(x)tan(x)
Thus, ddxsec(x)=sec(x)tan(x) //
6. Derivative of cosec(x):
Let, y=f(x)=cosec(x)=1sin(x)....(1)
Following the same process as in sec(x), you get,
ddxcosec(x)=-cosec(x)cot(x) //
Learn different rules of determining derivatives of different types of functions from HERE.
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