Answer to Question #136867 in Calculus for Millicent

Question

Find the derivative of sin(55)x and cos(164)x

Answer 1

$\displaystyle(1)\\ \textsf{Let} \hspace{0.1cm} y = \sin(55x) \hspace{0.1cm} \textsf{and}\hspace{0.1cm} u = 55x\\ \Rightarrow y = \sin{u}, \hspace{0.1cm}\&\hspace{0.1cm} \frac{\mathrm{d}y}{\mathrm{d}u} = \cos{u}\\ u = 55x, \frac{\mathrm{d}u}{\mathrm{d}x} = 55\\ \textsf{By Composite rule of differentiation,}\\ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}u} \times \frac{\mathrm{d}u}{\mathrm{d}x}\\ \therefore \frac{\mathrm{d}y}{\mathrm{d}x} = 55 \times \cos(u) = 55\cos{55x}\\ (2)\\ \textsf{Let} \hspace{0.1cm} y = \cos(164x) \hspace{0.1cm} \textsf{and}\hspace{0.1cm} v = 164x\\ \Rightarrow y = \cos{v}, \hspace{0.1cm}\&\hspace{0.1cm} \frac{\mathrm{d}y}{\mathrm{d}v} = -\sin{v}\\ v = 164x, \frac{\mathrm{d}v}{\mathrm{d}x} = 164\\ \textsf{By Composite rule of differentiation,}\\ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\mathrm{d}y}{\mathrm{d}v} \times \frac{\mathrm{d}v}{\mathrm{d}x}\\ \therefore \frac{\mathrm{d}y}{\mathrm{d}x} = 164 \times (-\sin(v)) = -164\sin{164x}$