Step 1: Start by putting [tex] \frac{d}{dx} [/tex] in front of each term
[tex] \frac{d}{dx}[y cos x]= \frac{d}{dx}[5x^2]+ \frac{d}{dx}[ 3y^2][/tex] ----------------------------------------------------------------------------------------------------------------- Step 2:
Deal with the terms in 'x' and the constant terms [tex] \frac{d}{dx}[ycosx]= 10x+ \frac{d}{dx} [3y^2] [/tex] ---------------------------------------------------------------------------------------------------------------- Step 3:
Use the chain rule for the terms in 'y' [tex] \frac{d}{dx}[ycosx]=10x+6y \frac{dy}{dx} [/tex] -------------------------------------------------------------------------------------------------------------- Step 4:
Use the product rule on the term in 'x' and 'y' [tex](y) \frac{d}{dx} cos x+(cos x) \frac{d}{dx}y =10x+6y \frac{dy}{dx}
[/tex] [tex]y(-siny)+(cosx) \frac{dy}{dx} =10x+6y \frac{dy}{dx} [/tex] --------------------------------------------------------------------------------------------------------------
Step 5:
Rearrange to make [tex] \frac{dy}{dx} [/tex] the subject [tex]-y sin(y)+cos(x) \frac{dy}{dx} =10x+6y \frac{dy}{dx} [/tex] [tex]cos(x) \frac{dy}{dx}-6y \frac{dy}{dx}=10x+y sin(y) [/tex] [tex][cos(x) - 6y] \frac{dy}{dx}=10x + y sin(y) [/tex] [tex] \frac{dy}{dx}= \frac{10x+ysin(y)}{cos(x)-6y} [/tex] ⇒ Final Answer
