Use implicit differentiation where $LaTeX: \displaystyle y$ is a function of $LaTeX: \displaystyle x$ to find the derivative of $LaTeX: \displaystyle 21 \sqrt{y} \cos{\left(x^{3} \right)} + 4 \log{\left(y \right)} \sin{\left(x^{2} \right)}=41$

Taking the derivative of both sides using implicit differentiation gives: $LaTeX: - 63 x^{2} \sqrt{y} \sin{\left(x^{3} \right)} + 8 x \log{\left(y \right)} \cos{\left(x^{2} \right)} + \frac{4 y' \sin{\left(x^{2} \right)}}{y} + \frac{21 y' \cos{\left(x^{3} \right)}}{2 \sqrt{y}} = 0$ Solving for $LaTeX: \displaystyle y'$ gives $LaTeX: \displaystyle y' = \frac{2 x \left(63 x y^{2} \sin{\left(x^{3} \right)} - 8 y^{\frac{3}{2}} \log{\left(y \right)} \cos{\left(x^{2} \right)}\right)}{8 \sqrt{y} \sin{\left(x^{2} \right)} + 21 y \cos{\left(x^{3} \right)}}$