Question

If $3\sin (xy) + 4\cos (xy) = 5$then $\dfrac{{dy}}{{dx}} =$
$A)\dfrac{{ - y}}{x}$
$B)\dfrac{y}{x}$
$C)\dfrac{x}{y}$
$D)$None of these

Answer 1

First, we need to analyze the given information which is in the trigonometric form.
The trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation and these identities are useful whenever expressions involving trigonometric functions need to be simplified.
Here we are asked to calculate the differentiation of $3\sin (xy) + 4\cos (xy) = 5$, so we need to know the formulas in sine and cos in the trigonometry.
In differentiation, the derivative of $x$ raised to the power is denoted by $\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$ .

Formula used:
$\dfrac{d}{{dx}}(\sin x) = \cos x$ and $\dfrac{d}{{dx}}\sin (xy) = \cos x\dfrac{d}{{dx}}(xy)$
$\dfrac{d}{{dx}}(\cos x) = - \sin x$ and $\dfrac{d}{{dx}}\cos (xy) = - \sin x\dfrac{d}{{dx}}(xy)$
$\dfrac{{\sin x}}{{\cos x}} = \tan x$
$\dfrac{d}{{dx}}(xy) = x\dfrac{{dy}}{{dx}} + y$ (By the $uv$ differentiation in the multiplication, which is $\dfrac{d}{{dx}}(xy) = x{y^1} + y{x^1} \Rightarrow x \times \dfrac{{dy}}{{dx}} + y \times 1$)

Complete step-by-step solution:
Since from the given that we have $3\sin (xy) + 4\cos (xy) = 5$and we need to differentiate with respect to the variable x.
In the process of differentiation, we only derive the variables but not constant and numbers even if it gets zero only.
Thus, starting with the differentiation of the given term on both sides, $3\sin (xy) + 4\cos (xy) = 5 \Rightarrow \dfrac{d}{{dx}}[3\sin (xy) + 4\cos (xy)] = \dfrac{d}{{dx}}(5)$
Thus, giving the derivative values to the variables, then we get $[3\dfrac{d}{{dx}}\sin (xy) + 4\dfrac{d}{{dx}}\cos (xy)] = \dfrac{d}{{dx}}(5)$
Since, as we said the derivative of the constant or a number is zero, then we get zero on the right side, $[3\dfrac{d}{{dx}}\sin (xy) + 4\dfrac{d}{{dx}}\cos (xy)] = \dfrac{d}{{dx}}(5) \Rightarrow [3\dfrac{d}{{dx}}\sin (xy) + 4\dfrac{d}{{dx}}\cos (xy)] = 0$
From the trigonometric formulas we have, $\dfrac{d}{{dx}}\sin (xy) = \cos x\dfrac{d}{{dx}}(xy)$and$\dfrac{d}{{dx}}\cos (xy) = - \sin x\dfrac{d}{{dx}}(xy)$
Hence, substituting the values to these functions we get, $[3\dfrac{d}{{dx}}\sin (xy) + 4\dfrac{d}{{dx}}\cos (xy)] = 0 \Rightarrow [3\cos (xy)\dfrac{d}{{dx}}(xy) - 4\sin (xy)\dfrac{d}{{dx}}(xy)] = 0$
Now, taking the common values out we get $[3\cos (xy) - 4\sin (xy)]\dfrac{d}{{dx}}(xy) = 0$
Again, by the formula $\dfrac{d}{{dx}}(xy) = x\dfrac{{dy}}{{dx}} + y$in the above equation, we get, $[3\cos (xy) - 4\sin (xy)](x\dfrac{{dy}}{{dx}} + y) = 0$
Since by the concept of multiplication, if $ab = 0$then either $a = 0$or $b = 0$
Applying this concept in the above we get, either $[3\cos (xy) - 4\sin (xy)] = 0$or $x\dfrac{{dy}}{{dx}} + y = 0$
Further solving both equations we get,
Either $[3\cos (xy) - 4\sin (xy)] = 0 \Rightarrow 3\cos (xy) = 4\sin (xy)$or $x\dfrac{{dy}}{{dx}} + y = 0 \Rightarrow x\dfrac{{dy}}{{dx}} = - y$
Either $3\cos (xy) = 4\sin (xy) \Rightarrow \dfrac{{\sin (xy)}}{{\cos (xy)}} = \dfrac{3}{4}$or $x\dfrac{{dy}}{{dx}} = - y \Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{{ - y}}{x}$
There we get, either $\tan (xy) = \dfrac{3}{4}$or $\dfrac{{dy}}{{dx}} = \dfrac{{ - y}}{x}$
But from the given question we only need to find the value of $\dfrac{{dy}}{{dx}}$and thus we get $\dfrac{{dy}}{{dx}} = \dfrac{{ - y}}{x}$
Hence, the option $A)\dfrac{{ - y}}{x}$is correct.

Note: Since we also have found the answer as $\tan (xy) = \dfrac{3}{4}$and if it is in the given options then it is also the correct option so don’t be confused about that.
The concept of either or is the values of one are correct and also the possibility that the second value will be correct like in the solution either $\tan (xy) = \dfrac{3}{4}$or $\dfrac{{dy}}{{dx}} = \dfrac{{ - y}}{x}$, both are correct.
Also, if the constant function is unique like in the solution $\dfrac{d}{{dx}}(5)$then we get zero, suppose the constant function is with the variables like $\dfrac{d}{{dx}}3\sin (xy)$ then we will ignore the constant and differentiate the variables only.