Question

Find 𝑑𝑦

𝑑𝑥

in each case

(a) 𝑦 = 𝑡𝑎𝑛4(3𝑥)

(3)

(b) 𝑥2𝑦2 + 𝑥𝑠𝑖𝑛𝑦 = 4 use implicit differentiation

Answer 1

Let us find $\frac{dy}{dx}$ in each case

(a) $𝑦 = \tan^4(3𝑥)$

(b) $𝑥^2𝑦^2 + 𝑥\sin𝑦 = 4$ use implicit differentiation

(a) $𝑦' = 4\tan^3(3𝑥)(\tan(3𝑥))'= 4\tan^3(3𝑥)\frac{1}{\cos^2 (3x)}(3𝑥)'= 12\frac{\tan^3(3𝑥)}{\cos^2 (3x)}.$

(b) Let us use the implicit differentiation:

$2x𝑦^2 + 𝑥^22𝑦y' +\sin𝑦+xy'\cos y = 0.$

Therefore, $y'(2𝑥^2y+x\cos y) = -2x𝑦^2 - \sin y,$ and hence

$y' =- \frac{2x𝑦^2 +\sin y}{2𝑥^2y+x\cos y}.$