Question

If $f(x,y)=\dfrac{\cos (x-4y)}{\cos (x+4y)}$, then at $(0,\dfrac{\pi }{4})$ is equal to
1. $-1$
2. $0$
3. $2$
4. $1$

Answer 1

In these types of questions, we have given a function and then we will partially differentiate that function and then we have to check which derivative rule can we apply and after that we have to substitute the values of given points in the partially differentiated function and we will get our required answer.

Complete step-by-step solution:
Differentiation can be used to measure the rate of change of the function or slope of a tangent line. We can take an example as the rate of change of displacement with respect to time, is called velocity. There are four partial derivative rules- Product Rule, Quotient Rule, Power Rule, Chain Rule
The product rule is the formula used to determine the derivative where one function is multiplied by another. Let suppose there are two functions: $f(x)$ and $g(x)$, then the product rule is as:
$(f.g)'=f'.g+f.g'$
The quotient rule is used to find the derivative of a function that can be written as the quotient of two functions. This rule can be stated as the (differentiation of first function multiplied by the second function) subtract by the (differentiation of second function multiplied by the first function) to the square of the second function. Quotient rule is as:
$[\dfrac{s(x)}{t(x)}]'=\dfrac{t(x).s'(x)-s(x).t'(x)}{{{\\{t(x)\\}}^{2}}}$
Chain rule can be used to find the derivative of the composition of two or more functions. The chain rule expresses the derivative of their composition:
$\dfrac{dy}{dx}=\dfrac{dy}{du}.\dfrac{du}{dx}$
Derivatives can also be used to find the maximum and minimum values of any given function.
As we have given in the question $f(x,y)=\dfrac{\cos (x-4y)}{\cos (x+4y)}$
So, $\dfrac{\partial f}{\partial x}=\dfrac{\partial }{\partial x}\left( \dfrac{\cos (x-4y)}{\cos (x+4y)} \right)$
After partial differentiation it becomes as,
$\dfrac{\partial f}{\partial x}=\dfrac{\left[ -\cos (x+4y)\sin (x-4y)+\cos (x-4y)\sin (x+4y) \right]}{{{\cos }^{2}}(x+y)}$
Now, we will use the formula $\sin (A+B)=\sin A\cos B+\cos A\sin B$.
Therefore by using this formula we will substitute the values. So it becomes as:
$\dfrac{\partial f}{\partial x}=\dfrac{\sin (x+4y-(x-4y))}{{{\cos }^{2}}(x+y)}$
$=\dfrac{\sin 8y}{{{\cos }^{2}}(x+y)}$
Now we will substitute the values of given point:
At $(0,\dfrac{\pi }{4})$,
$\dfrac{\partial f}{\partial x}=\dfrac{\sin \dfrac{8\pi }{4}}{{{\cos }^{2}}\dfrac{\pi }{4}}$
\Rightarrow $$$$\dfrac{\partial f}{\partial x}=\dfrac{\sin 2\pi }{{{\cos }^{2}}\dfrac{\pi }{4}}
Now we will put the values of $\sin 2\pi$ and ${{\cos }^{2}}\dfrac{\pi }{4}$, it becomes:
$\dfrac{\partial f}{\partial x}=0$
Hence, the correct option is $2$.

Note: Derivatives can be used in real life in different aspects like to calculate the profit and loss in business and also check the temperature variation. It is also used in calculating the approximate values of complex calculations. It is also used in finding the maximum and minimum values of any function.