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