﻿ Trigonometric Functions Derivatives Worksheet | Problems & Solutions

# Trigonometric Functions Derivatives Worksheet

Trigonometric Functions Derivatives Worksheet
• Page 1
1.
Find .
 a. tan 9$x$ b. $\frac{9}{2}$ tan $x$ c. $\frac{9}{2}$ tan 9$x$ d. tan 9$x$

#### Solution:

Dxsec 9x = Dx(sec 9x)12

= 12(sec 9x)-12Dx(sec 9x)
[Use the chain rule.]

= 12(sec 9x)-12(sec 9x tan 9x) 9

= 9 / 2 tan 9x sec 9x

2.
If $t$ = sin 4$x$ cos 4$x$, then find $\frac{dt}{dx}$.
 a. sin 8$x$ b. cos 8$x$ c. cos $x$ d. 4cos 8$x$

#### Solution:

t = (sin 4x)(cos 4x) = 1 / 2 (sin 8x)
[Use sinθcosθ
= (1 / 2)sin2θ.]

Hence, dtdx = 1 / 2(8cos 8x) = 4cos 8x
[Use the chain rule.]

3.
If $y$ = , then find $\frac{dy}{dx}$.
 a. - b. - c. - d. -

#### Solution:

dydx = (21+sin x)Dx (8cos x) - 8cos xDx (21+sin x)(21+sin x)2
[Use the Quotient Rule.]

= (21 + sin x)(- 8sin x) - 8cos x(cos x)(21+sin x)2

= - 168sin x-8sin2 x-8cos2 x(21+sin x)2

= - 8(1+21sin x)(21+sin x)2
[Use sin² x + cos² x = 1.]

dydx = - 8(1+21sin x)(21+sin x)2

4.
If $f$ ($t$) = , then find $f$ ′($t$).
 a. 5 b. c. 5 d.

#### Solution:

f ′(t) = 52(18 + 2sin t)32Dt(18 + 2sin t)
[Use the chain rule.]

= 52(18 + 2sin t)32 (2cos t)

= 5 (18 + 2sin t)32 (cos t)

5.
Find .
 a. 7(sec (sin 3$x$) tan(sin 3$x$)) cos 3$x$ b. (sec (sin 3$x$) tan(sin 3$x$)) cos 3$x$ c. 21(sec (sin 3$x$) tan(sin 3$x$)) cos 3$x$ d. 21(sec (sin 3$x$) tan(sin 3$x$)) cos $x$

#### Solution:

ddx[7sec (sin 3x)] = 7sec (sin 3x) tan (sin 3x)[Dx sin 3x]
[Use the Chain Rule.]

= 21(sec (sin 3x) tan(sin 3x)) cos 3x

6.
Find D$x$(sin2 4$x$ sin4 6$x$).
 a. 4sin 8$x$ sin4 6$x$ + 24sin2 4$x$ sin3 6$x$ cos 6$x$ b. 4sin 8$x$ sin4 6$x$ + 24sin2 4$x$ sin3 6$x$ c. 4sin 8$x$ sin4 6$x$ + 6sin2 4$x$ sin3 6$x$ cos 6$x$ d. 4sin 8$x$ sin4 + 24sin2 4$x$ sin3 6$x$ cos 6$x$

#### Solution:

Dx(sin2 4x sin4 6x)

= [Dx (sin2 4x)](sin4 6x) + (sin2 4x)[Dx (sin4 6x)]
[Use the product rule.]

= (2 sin 4x) (Dx (sin 4x)) (sin4 6x) + (sin2 4x) (4sin3 6x)(Dx (sin 6x))

= (2sin 4x)(4 cos 4x) (sin4 6x) + (sin2 4x) (4sin3 6x)(6 cos 6x)

= 8sin 4x cos 4x sin4 6x + 24 sin2 4x sin3 6x cos 6x

= 4sin 8x sin4 6x + 24 sin2 4x sin3 6x cos 6x
[Use 2 sin θ cos θ = sin 2θ.]

7.
Differentiate $f$($x$) = tan ${x}^{\frac{7}{2}}$.
 a. ) b. ) c. (${\mathrm{sec}}^{2}{x}^{\frac{7}{2}}$) d. )

#### Solution:

f ′(x) = ddx (tanx72)

= (sec2 x72)ddx (x72)
[Use the chain rule.]

= (sec2 x72) 7 / 2 (x52)

= 7x522 (sec2x72)

8.
If $y$ = , then find $\frac{dy}{dx}$.
 a. - 14 b. - [sin 2${\left(4x+34\right)}^{\frac{5}{2}}\right]$ c. - ${\left(4x+34\right)}^{\frac{5}{2}}$ d. - 4

#### Solution:

dydx = 2[cos(4x+34)72] Dx [cos (4x+34)72]
[Use the chain rule.]

= [2cos (4x+34)72] [- sin(4x+34)72]Dx(4x+34)72
[Use the chain rule again.]

= - 2[sin (4x+34)72] [cos(4x+34)72] (72)(4x+34)52(4)

= - 14 (4x+34)52 [sin 2(4x+34)72]
[Use 2 sin θ cos θ = sin 2θ.]

9.
If $y$ = sin 7$x$ cos 9$x$, then find $\frac{dy}{dx}$.
 a. - 9 sin 7$x$ sin 9$x$ + 7 cos 7$x$ cos 9$x$ b. - 7sin 7$x$ sin 9$x$ + 9 cos 7$x$ cos 9$x$ c. - sin 7$x$ sin 9$x$ + cos 7$x$ cos 9$x$ d. 9 sin 7$x$ sin 9$x$ - 7 cos 7$x$ cos 9$x$

#### Solution:

dydx =ddx(sin 7x cos 9x) = sin 7xddx(cos 9x) + cos 9xddx(sin 7x)
[Use the Product Rule.]

= sin 7x [(- sin 9x)9] + cos 9x [(cos 7x)7]

= - 9 sin 7x sin 9x + 7 cos 7x cos 9x

10.
If $y$ = , then find $\frac{dy}{dx}$.
 a. ) b. does not exist c. d.

#### Solution:

dydx = cos2 x+sin2 3xDx (5)-Dx (cos2 x+sin2 3x)cos2 x+sin2 3x
[Use the quotient rule.]

= - 52cos2 x+sin2 3x2 cosx(- sinx)+2sin 3x(cos 3x)(3)cos2 x+sin2 3x

= - 52cos2 x+sin2 3x(3sin 6x-sin 2xcos2 x+sin2 3x)
[Use 2sin θ cos θ = sin2 θ.]

= 52cos2 x+sin2 3x(sin 2x - 3sin 6xcos2 x+sin2 3x)