Question

How do you differentiate \[\sin x+\cos y=\sin x\cos y\]?

Answer 1

Hint: Differentiation refers to the rate with which a function changes. So change of that function is found using another parameter with which the change is measured. So we will differentiate y terms with respect to x and solve \[\dfrac{dy}{dx}\].
For eg – if we want to differentiate sin x with respect to x, we get,
\[\sin x=\dfrac{d}{dx}(\sin x)=\cos x\]

Complete step by step answer:
According to the question we have to differentiate \[\sin x+\cos y=\sin x\cos y\].
Before starting to differentiate, check what is asked in the question whether it is \[\dfrac{dy}{dx}\] or \[\dfrac{dx}{dy}\]. At first it might hardly seem any different but it is what makes all the difference.
\[\dfrac{dy}{dx}\] means that we have to differentiate \[y\] with respect to \[x\].
Similarly, \[\dfrac{dx}{dy}\] means that we have to differentiate \[x\] with respect to \[y\].
So, in the given question we are asked to differentiate \[y\] with respect to \[x\], that is, \[\dfrac{dy}{dx}\].
Differentiating both sides, we get
\[\dfrac{d}{dx}(\sin x+\cos y)=\dfrac{d}{dx}(\sin x\cos y)\]
\[\dfrac{d}{dx}(\sin x)+\dfrac{d}{dx}(\cos y)=\dfrac{d}{dx}(\sin x\cos y)\]
We know that, derivative of \[\sin x\] is \[\cos x\]. And the derivative \[\cos y\] will be different, \[\cos y\] will be first differentiated and since the variable is y, so the differentiation of y will be \[\dfrac{dy}{dx}\].
On the right hand side of the equation, we have \[\sin x\cos y\], so it will undergo differentiation using the product rule.
We get,
\[\cos x+(-\sin y)\dfrac{dy}{dx}=\sin x\dfrac{d}{dx}(\cos y)+\cos y\dfrac{d}{dx}(\sin x)\]
\[\cos x-\sin y\dfrac{dy}{dx}=\sin x(-\sin y)\dfrac{dy}{dx}+\cos y(\cos x)\]
Now, we will rearrange the equation to get \[\dfrac{dy}{dx}\] on one side and the rest on the other side.
\[\sin x(\sin y)\dfrac{dy}{dx}-\sin y\dfrac{dy}{dx}=\cos y(\cos x)-\cos x\]
\[(\sin x(\sin y)-\sin y)\dfrac{dy}{dx}=\cos y(\cos x)-\cos x\]
\[\Rightarrow \dfrac{dy}{dx}=\dfrac{\cos y(\cos x)-\cos x}{(\sin x(\sin y)-\sin y)}=\dfrac{\cos x(\cos y-1)}{\sin y(\sin x-1)}\]
Therefore, \[\dfrac{dy}{dx}=\dfrac{\cos x(\cos y-1)}{\sin y(\sin x-1)}\]

Note:
While solving the question care should be given to the parameter with which the rate of change of the function is asked else it will result in a wrong answer. The differentiation should be carried out step wise else if a step is missed then the entire answer will go wrong.