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Find the derivatives of Log(secx) by using first principle

Dragon Gamer , 3 Years ago
Grade 12
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Yogendra Kumar

Last Activity: 3 Years ago

To find the derivative of the function y=log(secx) using the first principle of derivatives, we'll rely on the definition of the derivative as a limit. The first principle states that the derivative of a function f(x) at a point x is given by:

Defining the Derivative

The derivative f(x) can be expressed mathematically as:

f(x)=limh0f(x+h)f(x)h

Applying the Definition

Let's apply this to our function y=log(secx). First, we need to find f(x+h):

So, we have:

\( f(x + h) = \log(\sec(x + h))

Now, we can substitute f(x) and f(x+h) into the limit definition:

f(x)=limh0log(sec(x+h))log(secx)h

Using Logarithmic Properties

We can utilize the properties of logarithms, specifically that log(a)log(b)=log(ab), to simplify the expression:

\( f'(x) = \lim_{h \to 0} \frac{1}{h} \log\left(\frac{\sec(x + h)}{\sec x}\right)

Limit Transformation

Next, we can rewrite this limit. We know that sec(x)=1cos(x), so:

\( \frac{\sec(x + h)}{\sec x} = \frac{\cos x}{\cos(x + h)}

This gives us:

\( f'(x) = \lim_{h \to 0} \frac{1}{h} \log\left(\frac{\cos x}{\cos(x + h)}\right)

Using Taylor Series Expansion

To find the limit, we can use the Taylor series expansion for cos(x+h) around h=0:

  • cos(x+h)cosxhsinx+O(h2)

Substituting this approximation into our expression, we get:

cosxcos(x+h)cosxcosxhsinx=11hsinxcosx

Utilizing the limit property log(1+u)u for small u, we can write:

\( \log\left(\frac{\cos x}{\cos(x + h)}\right) \approx \frac{h \sin x}{\cos x}

Final Steps to Derivative

Now we substitute back into our derivative limit:

f'(x) = \lim_{h \to 0} \frac{1}{h} \cdot \frac{h \sin x}{\cos x} = \frac{\sin x}{\cos x} = \tan x

Conclusion

Thus, we have derived that:

\( \frac{d}{dx} \log(\sec x) = \tan x

This process not only showcases the power of the first principles in calculus but also reinforces our understanding of how derivatives represent the rate of change of functions in a foundational way.

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