Question

Find the dy/dx for the equation below using implicit differentiation.

Answer 1

First of all, recall that y is a product of function of x. We have three factors:

$f(x)=x^2,\quad g(x)=e^{3x},\quad h(x)=\sin(4x)$

The derivative of a product of function is computed by deriving one function at the time, and then adding all the results:

$y' = f'(x)g(x)h(x)+f(x)g'(x)h(x)+f(x)g(x)h'(x)$

Let's compute the derivative of each function first:

$f'(x)=2x,\quad g'(x) = 3e^{3x},\quad h'(x)=4\cos(4x)$

Now plug f, f', g, g', h, h' in the formula above as required:

$2xe^{3x}\sin(4x)+3x^2e^{3x}\sin(4x)+4x^2e^{3x}\cos(4x)$