University:
York University
Course:
MATH 1013 | Applied Calculus I
Academic year:

2022
Views:

473

Pages:

4
Author:

Nick T.

3.3 Derivatives of Trig Functions f(x) = sin x f'(x) = cos x f'(x) = lim(h->0) [f(x+h) - f(x)] / h = lim(h->0) [sin(x+h) - sin x] / h lim(h->0) [sin x cos h + cos x sin h - sin x] / h = lim(h->0) [sin x (cos h - 1) + cos x (sin h)] / h = lim(h->0) sin x * lim(h->0) (cos h - 1) / h + lim(h->0) cos x * lim(h->0) (sin h) / h = sin x * 0 + cos x * 1 = cos x lim(h->0) sin x = sin x lim(h->0) cos x = cos x lim(h->0) (sin h) / h = 1 lim(h->0) (cos h - 1) / h = 0 Formula for derivative sine function: d/dx(sin x) = cos x Derivatives of Trigonometric Functions d/dx(sin(x)) = cos(x) d/dx(cos(x)) = -sin(x) d/dx(tan(x)) = sec^2(x) d/dx(cot(x)) = -csc^2(x) d/dx(sec(x)) = sec(x)tan(x) d/dx(csc(x)) = -csc(x)cot(x) y = x^2 sin(x) (x^2)(cos(x)) + (sin(x))(2x) (x^2)cos(x) + 2x sin(x) cos^2(x) + sin^2(x) = 1 cos^2(x) - sin^2(x) = cos(2x) (cos(x) + sin(x))^2 - 2sin(x)cos(x) = 1 + sin(2x) cos^2(x) - sin^2(x) / cos^2(x) = 1 + tan^2(x) sin(θ) = π/10 10sin(θ) = π d(10sin(θ))/dθ = dπ/dθ 10cos(θ) = 0 How fast does x change with respect to θ? dθ/dx = 1 / (10cos(θ)) dθ/dx | θ=π/3 = 1 / (10cos(π/3)) = 1/(10*(1/2)) = 1/5