Question

Find the derivatives of each of the following, showing the working and simplifying.
y = y(x), where x3 sin y + cos(2x + 5y) = 1

Answer 1

$$x ^ {3} + \sin y + \cos (2 x + 5 y) = 1$$

We differentiate both sides of the equation like a composite function by $x$:

$$3 x ^ {2} \sin y + x ^ {3} \cos y y ^ {\prime} - \sin (2 x + 5 y) (2 + 5 y ^ {\prime}) = 0$$ $$3 x ^ {2} \sin y + x ^ {3} \cos y y ^ {\prime} - 2 \sin (2 x + 5 y) - 5 y ^ {\prime} \sin (2 x + 5 y) = 0$$ $$y ^ {\prime} \left(x ^ {3} \cos y - 5 \sin (2 x + 5 y)\right) = 2 \sin (2 x + 5 y) - x ^ {2} \sin y$$ $$y ^ {\prime} = \frac {2 \sin (2 x + 5 y) - x ^ {2} \sin y}{x ^ {3} \cos y - 5 \sin (2 x + 5 y)}$$