Question

# Find $f'\left( x \right)$.$f\left( x \right)=\sec x-\sqrt{2}\tan x$

Hint: To solve the above problem first we have to find the basic derivatives of $\sec x$ and $\tan x$. After substituting the derivatives in the equation, rewrite the equation with the derivatives of the function. Solve the equation to find the final answer.

$f'\left( x \right)=\dfrac{d}{dx}\left( \sec x \right)-\dfrac{d}{dx}\left( \sqrt{2}\tan x \right)$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
We know the derivative of $\sec x$ is $\sec x\cdot \tan x$ and the derivative of $\tan x$ is ${{\sec }^{2}}x$.
On substituting the derivatives of $\sec x$ and $\tan x$ in the above equation we get,
$f'\left( x \right)=\sec x\cdot \tan x-\sqrt{2}{{\sec }^{2}}x$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
Taking $\sec x$ as common in the right hand side (RHS) we get,
$f'\left( x \right)=\sec x\left( \tan x-\sqrt{2}\sec x \right)$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
Hence the value of $f'\left( x \right)$ is $\sec x\left( \tan x-\sqrt{2}\sec x \right)$.
Note: The possible error that you may encounter can be the wrong substitution values of the derivatives of $\sec x$ and $\tan x$. Solving the equation should be done carefully. It is to note here that integers are exempted from the calculation of derivatives.