Question

What is the derivative of $\tan xy?$

Answer 1

We will use the usual rule of differentiation of functions. If we need to differentiate a function $y=f\left( u \right)$ with respect to $x,$ then we will first differentiate the whole of the function with respect to $u$ and then we will differentiate $u$ with respect to $x.$ That can be mathematically expressed as $\dfrac{df\left( u \right)}{dx}=\dfrac{df}{du}\dfrac{du}{dx}.$

Complete step-by-step solution:
Let us consider the given trigonometric function $\tan xy.$
We are asked to find the derivative of the given trigonometric function.
We know that if $u$ is a function of $x$ and $f$ is a function of $u$ and if we are asked to find the derivative of the function $f$ with respect to $x,$ then we will have to differentiate $f$ with respect to $u$ and then multiply the derivative with the derivative of $u$ with respect to $x.$
And we can express it mathematically as $\dfrac{df\left( u \right)}{dx}=\dfrac{df}{du}\dfrac{du}{dx}.$
Now, let us compare the given function with the above identity.
Then, we will get $f\left( u \right)=\tan xy$ and $u=xy.$
When we differentiate the given function with the help of the above identity, we will get the left-hand side of the equation as $\dfrac{df\left( u \right)}{dx}=\dfrac{d\tan xy}{dx}.$
Now let us take $u=xy$ and we will get $\dfrac{df}{dx}=\dfrac{d}{dx}\tan u$ and $\dfrac{du}{dx}=\dfrac{dxy}{dx}.$
We know that the derivative of $\tan u$ with respect to $u$ is ${{\sec }^{2}}u$ and the derivative of $xy$ with respect to $x$ is $y.$
That is, we will get $\dfrac{d}{dx}\tan u={{\sec }^{2}}u$ and $\dfrac{dxy}{dx}=y.$
Therefore, we will get the right-hand side of the identity as $\dfrac{df}{du}\dfrac{du}{dx}=\dfrac{d}{du}\tan u\dfrac{du}{dx}=\dfrac{d}{du}\tan u\dfrac{dxy}{dx}.$
Now, we will get $\dfrac{df}{du}\dfrac{du}{dx}=\dfrac{d}{du}\tan u\dfrac{du}{dx}={{\sec }^{2}}u\cdot y.$
And that is. $\dfrac{df}{du}\dfrac{du}{dx}=y{{\sec }^{2}}xy.$
Hence the derivative of the given trigonometric function is $y{{\sec }^{2}}xy.$

Note: When we differentiate a function, we should always remember the identity $\dfrac{df\left( u \right)}{dx}=\dfrac{df}{du}\dfrac{du}{dx}.$ We sometimes make a mistake by forgetting the part $\dfrac{du}{dx}$ in the identity and it leads us to the wrong answer.