Question

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

Answer 1

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.

Complete step-by-step answer:
Applying derivative on both sides of the equation with respect to x we get,
$$f^{\prime }\left(x\right)={\displaystyle \frac{d}{dx}}(\sec x)-{\displaystyle \frac{d}{dx}}(\sqrt{2}\tan x)$$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (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,
$$f^{\prime }\left(x\right)=\sec x\cdot \tan x-\sqrt{2}{\sec }^{2}x$$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
Taking $$\sec x$$ as common in the right hand side (RHS) we get,
$$f^{\prime }\left(x\right)=\sec x(\tan x-\sqrt{2}\sec x)$$. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
Hence the value of $$f^{\prime }\left(x\right)$$ is $$\sec x(\tan x-\sqrt{2}\sec x)$$.

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.