Derivative of sin x and cos x.

We know the formula for finding the derivative
dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0

Derivative of sin x
y = f(x) = sin x
f(x + δx) = sin (x + δx) = sin x cos δx + cos x sin δx

Difference
f(x + δx) - f(x) = sin (x + δx) - sin x
= sin x cos δx + cos x sin δx - sin x
= sin x (cos δx - 1) + cos x sin δx

The ratio
[f(x + δx) - f(x)]/δx = sin x (cos δx - 1)/δx + cos x (sin δx)/δx
 
dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0

When δx→0 then (cos δx - 1)/δx → 0 and (sin δx)/δx → 1

Hence the value of dy/dx = d(sin x)/dx = sin x × 0 + cos x × 1 = cos x.

Derivative of cos x
y = f(x) = cos x
f(x + δx) = cos (x + δx) = cos x cos δx - sin x sin δx

Difference
f(x + δx) - f(x) = cos (x + δx) - cos x
= cos x cos δx - sin x sin δx - cos x
= cos x (cos δx - 1) - sin x sin δx

The ratio
[f(x + δx) - f(x)]/δx = cos x (cos δx - 1)/δx + sin x (sin δx)/δx
 
dy/dx = lim [f(x + δx) - f(x)]/δx
       δx→0

When δx→0 then (cos δx - 1)/δx → 0 and (sin δx)/δx → 1

Hence the value of dy/dx = d(cos x)/dx = cos x × 0 - sin x × 1 = -sin x.

d(sin x)/dx = cos x
and
d(cos x)/dx = - sin x



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