what is the derivative of y= sin^4(5theta)?

Answer 1

Use the chain rule:

sin^4 (5Θ)

Differentiate the fourth power:

4sin³(5Θ)

Differentiate the sin(5Θ):

4sin³(5Θ)cos(5Θ)

Differentiate 5Θ

4sin³(5Θ)cos(5Θ)(5)

Simplify:

20sin³(5Θ)cos(5Θ)

Answer 2

y = sin^4(5 theta). Power rule: d u^n = n u^(n-1) du, so

dy/d theta = 4 sin^3(5 theta) d (sin(5 theta)), by chain rule. d sin(u) = cos u du, so

= 4 sin^3(5 theta) cos (5 theta) d (5 theta), chain rule
= 4 sin^3(theta) cos (5 theta) 5 d theta/d theta, so you stop, or
= 20 sin^3(theta) cos (5 theta).

Answer 3

20 sin^3 (5 theta) cos(5*theta)

Answer 4

y = sin^4 (5θ)
so dy/dθ = 4 sin^3 (5θ) d/dθ sin (5θ)
= 4 sin^3 (5θ) cos (5θ) . 5
= 20 sin^3 (5θ) cos (5θ)