FIND THE DERIVATIVE OF TRIGONOMETRIC FUNCTIONS

Problem 1 :

√x tan x

Solution :

= d/dx [√x tan x]

Problem 2 :

√(x tan x)

Solution :

= d/dx [√(x tan x)]

Problem 3 :

sin 2x + cos 3x

Solution : 

= d/dx [sin 2x + cos 3x]

= cos 2x · d/dx (2x( + (- sin 3x) · d/dx [3x]

= cos 2x (2) - sin 3x (3)

dy/dx = 2 cos 2x - 3 sin 3x

Problem 4 :

sin (4θ + π/2)

Solution :

= sin (4θ + π/2)

y = cos 4θ

dy/dθ = d/dθ [cos (4θ)]

dy/dθ = (-sin 4θ) · d/dθ [4θ]

dy/dθ = -4 sin (4θ)

Problem 5 :

tan θ + sec θ

Solution :

= d/dx [tan θ + sec θ]

= d/dθ [tan θ] + d/dθ [sec θ]

= sec² (θ) + sec (θ) tan (θ)

Factoring sec θ

= sec (θ) [tan (θ) + sec (θ)]

Problem 6 :

sin 2x cos 3x

Solution :

= d/dx [sin 2x cos 3x]

Since two differentiable functions are multiplied, we can find the derivative using product rule.

u = sin 2x ==> u' = cos 2x (2) ==> u' = 2 cos 2x

v = cos 3x ==> v' = - sin 3x (3) ==> u' = -3 sin 3x

= 2 cos (2x) cos (3x) - 3 sin (2x) sin (3x)

Problem 7 :

sin (2 cos 3x)

Solution :   

= d/dx [sin (2 cos 3x)]

= cos (2 cos (3x)) · d/dx [2 cos (3x)]

= cos (2 cos (3x)) · 2 (-sin 3x) (3)

= - 6 cos (2 cos (3x)) (sin 3x)

= -6 sin (3x) cos (2 cos (3x))

Problem 8 :

sec³ x⁴

Solution : 

= d/dx [sec³ x⁴]

= 3 sec² (x⁴) sec (x⁴) tan (x⁴) d/dx [x⁴]

= 3 tan (x⁴) 4 sec³ (x⁴) x³

= 12x³ sec³ (x⁴) tan (x⁴)

Problem 9 :

arctan (3x - 5)

Solution :

y = tan^-1(3x - 5)

Problem 10 :

arcsin (3x - 5)

Solution :   

y = sin^-1(3x - 5)

Problem 11 :

2^x+cosx

Solution :   

y = 2^x+cosx

d(a^x)/dx = a^x ln a

= d/dx [2^x+cosx]

= ln (2) · 2^x+cosx · d/dx [x + cos x]

= ln (2) · 2^x+cosx · d/dx [x] + d/dx [cos x]

= ln (2) · 2^x+cosx · (1 - sin x)

Problem 12 :

y = x log x

Solution :

y = x log x

Since two differentiable functions are multiplied, we use the product rule.

u = x and v = log x

u' = 1 and v' = 1/x

dy/dx = x (1/x) + log x (1)

dy/dx = 1 + log x

Problem 13 :

y = (sin x)²

Solution :   

= d/dx [(sin x)²]

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

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

dy/dx = sin 2x 

Problem 14 :

y = sin 2x - cos 4x

Solution :

= d/dx [sin 2x - cos 4x]

= d/dx [sin 2x] - d/dx [cos 4x]

= cos 2x d/dx [2x] - (-sin 4x) d/dx [4x]

= cos (2x) (2) + sin 4x (4)

= 2 cos (2x) + 4 sin 4x

= 4 sin 4x + 2 cos 2x

Problem 15:

y = 4x sin x

Solution :   

= d/dx [4x sin(x)]

= 4 d/dx [x sin(x)]

u = x and v = sin x

u' = 1 and v' = cos x

= 4 [ x cos x + sin x (1) ]

= 4x cos x + 4 sin x

Problem 16 :

y = log(x² + 1)

Solution :

= d/dx [log(x² + 1)]

= 1/(x² + 1) · d/dx [x² + 1]

= 2x + 0 / (x² + 1)

= 2x / (x² + 1)

Problem 17:

y = e^x sin x

Solution:   

= d/dx [e^x sin x]

= d/dx [e^x] sin (x) + e^x · d/dx [sin (x)]

= e^x sin (x) + e^x cos (x)

= e^x [sin (x) + cos (x)]