Find \[f'\left( x \right)\].
\[f\left( x \right)=\sec x-\sqrt{2}\tan x\]
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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,
\[f'\left( x \right)=\dfrac{d}{dx}\left( \sec x \right)-\dfrac{d}{dx}\left( \sqrt{2}\tan x \right)\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (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.
Complete step-by-step answer:
Applying derivative on both sides of the equation with respect to x we get,
\[f'\left( x \right)=\dfrac{d}{dx}\left( \sec x \right)-\dfrac{d}{dx}\left( \sqrt{2}\tan x \right)\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (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.