How do you differentiate #f(x)=(tanx-1)/secx# at #x=pi/3#?

Answer 1

#(1 + sqrt(3))/2#

Use sine and cosine to rewrite.

#f(x) = (sinx/cosx- 1)/(1/cosx)#
#f(x) = ((sinx - cosx)/cosx)/(1/cosx)#
#f(x) = sinx - cosx#
We differentiate this using #d/dx(sinx) = cosx# and #d/dx(cosx) = -sinx#.
#f'(x) = cosx - (-sinx)#
#f'(x) = cosx + sinx#
We now evaluate #f'(pi/3)#:
#f'(pi/3) = cos(pi/3) + sin(pi/3)#
#f'(pi/3) = 1/2 + sqrt(3)/2#
#f'(pi/3) = (1 + sqrt(3))/2#

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Answer 2

To differentiate ( f(x) = \frac{\tan(x) - 1}{\sec(x)} ) at ( x = \frac{\pi}{3} ), you can use the quotient rule and trigonometric identities.

First, we rewrite the function using trigonometric identities: [ f(x) = (\tan(x) - 1) \cos(x) ]

Then, differentiate ( f(x) ) using the product rule: [ f'(x) = (\tan(x) - 1)' \cos(x) + (\tan(x) - 1) (-\sin(x)) ]

Simplify by differentiating ( \tan(x) - 1 ): [ f'(x) = (\sec^2(x) - 0) \cos(x) + (\tan(x) - 1) (-\sin(x)) ] [ f'(x) = \sec^2(x) \cos(x) - \sin(x)(\tan(x) - 1) ]

Now, evaluate ( f'(x) ) at ( x = \frac{\pi}{3} ): [ f'\left(\frac{\pi}{3}\right) = \sec^2\left(\frac{\pi}{3}\right) \cos\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{3}\right)(\tan\left(\frac{\pi}{3}\right) - 1) ]

Using trigonometric values for ( \frac{\pi}{3} ): [ \sec\left(\frac{\pi}{3}\right) = 2, \quad \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \quad \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}, \quad \tan\left(\frac{\pi}{3}\right) = \sqrt{3} ]

Substitute these values into the expression for ( f'\left(\frac{\pi}{3}\right) ): [ f'\left(\frac{\pi}{3}\right) = (2^2) \left(\frac{1}{2}\right) - \left(\frac{\sqrt{3}}{2}\right)(\sqrt{3} - 1) ] [ f'\left(\frac{\pi}{3}\right) = 4 \cdot \frac{1}{2} - \frac{3}{2} + \frac{\sqrt{3}}{2} ] [ f'\left(\frac{\pi}{3}\right) = 2 - \frac{3}{2} + \frac{\sqrt{3}}{2} ] [ f'\left(\frac{\pi}{3}\right) = \frac{1}{2} + \frac{\sqrt{3}}{2} ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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