FINDING THE DERIVATIVE OF TRIG FUNCTIONS

Problem 1 :

y = cos³x

Solution :

y = cos³x

y = d/dx (cos³x)

= 3cos²(x) · (-sin x)

dy/dx = -3 sin x cos² x

Problem 2 :

y = x³ tan 8x

Solution :

y = x³ tan 8x

Here two differentiable functions are multiplied. So, we have to use product rule to differentiate the function given above.

= d/dx (x³ tan 8x)

u = x³ and v = tan 8x

u' = 3x² an dv' = 8sec² (8x)

d(uv) = uv' + vu'

= 3x² tan 8x + x³ sec² (8x) (8)

dy/dx = 3x² tan 8x + 8x³ sec² (8x)

Problem 3 :

y = sin² 4x

Solution :

y = sin² 4x

dy/dx = 2sin (4x) · d/dx(sin 4x)

= 2 sin (4x) cos (4x) · d/dx (4x)

= 2 sin (4x) cos (4x) · (4)

dy/dx = 4 sin 2(4x)

dy/dx = 4 sin 8x

Problem 4 :

y = cos (x² + 1)

Solution :

y = cos (x² + 1)

y = d/dx (cos (x² + 1))

dy/dx = - sin(x² + 1) d/dx (x² + 1)

= - (2x + 0) sin (x² + 1)

dy/dx = -2x sin (x² + 1)

Problem 5 :

y = tan πx

Solution :

y = tan πx

y = d/dx (tan πx)

dy/dx = sec² (πx) d/dx (πx)

= sec² (πx) (π)

dy/dx = π sec² (πx)

Problem 6 :

y = cos √x

Solution :

y = cos √x

y = d/dx (cos √x)

dy/dx = (- sin (√x) . d/dx (√x))

= -1/2 x^1/2-1 sin (√x)

dy/dx = - sin (√x) / 2√x

Problem 7 :

y = √cos 2x

Solution :

y = √cos 2x

y = d/dx (√cos 2x)

dy/dx = (1/(2√cos 2x))  sin 2x (2)

dy/dx = (2sin 2x/(2√cos 2x))

dy/dx = (sin 2x/(√cos 2x))

Problem 8 :

y = sin⁴ √x

Solution :

y = sin⁴ √x

y = d/dx (sin⁴ √x)

dy/dx = 4 sin³ (√x)  cos√x (1/2√x)

dy/dx = (4 sin³ (√x)  cos√x) / 2√x

dy/dx = (2 sin³ (√x)  cos√x) / √x

dy/dx = (sin² (√x))  (2 sin (√x)  cos√x) / √x

dy/dx = (sin² (√x))  (sin (2√x)) / √x

Problem 9 :

y = tan³x

Solution :

y = tan³x

y = d/dx (tan³x)

= 3tan²(x) · d/dx (tan x)

dy/dx = 3 tan² x sec² x

Problem 10 :

y = sin²x + cos²x

Solution :

y = sin²x + cos²x

y = dy/dx (sin²x + cos²x)

dy/dx = 2 sin x · d/dx (sin x) + 2 cos x · d/dx (cos x)

= 2 sin x cos x + 2 cos x (-sin x)

dy/dx = 0

Problem 11 :

y = (sin x + cos x)²

Solution :

y = (sin x + cos x)²

y = d/dx (sin x + cos x)²

dy/dx = 2 (sin x + cos x) · d/dx [(sin x) + (cos x)]

= 2 (sin x + cos x) (cos x - sin x)

dy/dx = 2 (cos² x - sin² x)

Using the formula, cos 2x = cos² x - sin² x

dy/dx = 2 (cos 2x)