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What is the derivative of secx?

Answer
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Hint: In this question we have been asked to find the derivative of the given trigonometric function secx. We will first rewrite the expression in the form of cosx and then we will use the formula of the derivative of the term in the form of uv. We will use the formula ddxuv=vdudxudvdxv2 and simplify the terms to get the required solution.

Complete step-by-step solution:
We have the term given to us as:
secx
Since we have to find the derivative of the term, it can be written as:
ddxsecx
Now we know that secx=1cosx therefore, on substituting, we get:
ddx1cosx
We can see that the expression is in the form of the derivative of uv.
On using the formula ddxuv=vdudxudvdxv2 on the expression, we get:
cosxddx11ddxcosxcos2x
Now we know that ddxk=0, where k is any constant value and ddxcosx=sinx therefore on substituting them in the expression, we get:
cosx(0)1(sinx)cos2x
On simplifying the terms, we get:
sinxcos2x
Now the denominator can be split up and written as:
sinxcosx×cosx
Now we know that sinxcosx=tanx, therefore on substituting, we get:
tanxcosx
Now we know that 1cosx=secx therefore, on substituting, we get:
secxtanx, which is the required derivative.
Therefore, we can write:
ddxsecx=secxtanx

Note: It is to be remembered that the function we used to solve the expression is called as the quotient rule. There also exists another rule which is known as the product rule which deals with expressions in the form of uv and has formula ddxuv=udvdx+vdudx. It is to be noted that the terms u and v are also written as f(x) and g(x) in some solutions.

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