How do you find the derivative of #f(x)=(x^3-8)tan^2(5x-3)#?

Answer 1

#f'(x)= 3x^2 tan^2 (5x-3) +10(x^3 -8) tan (5x-3) sec^2 (5x-3)#

First apply product rule of differentiation, #f' (x)= tan^2 (5x-3) d/dx (x^3 -8) + (x^3 -8) d/dx tan^2 (5x-3)#
=#3x^2 tan^2 (5x-3) +(x^3 -8) d/dx tan^2 (5x-3)#
Now apply chain rule to differentiate #tan^2 (5x-3)#. Let 5x-3=t, so that #d/dx tan^2 (5x-3)= d/dt tan^2 t dt/dx#
= #2 tan t sec^2 t d/dx (5x-3)#
= #2 tan (5x-3) sec^2 (5x-3) (5)#
=#10 tan (5x-3) sec^2 (5x-3)#
Then #f'(x)= 3x^2 tan^2 (5x-3) +10(x^3 -8) tan (5x-3) sec^2 (5x-3)#
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Answer 2

To find the derivative of ( f(x) = (x^3 - 8)\tan^2(5x - 3) ), you can use the product rule and the chain rule. Here's the step-by-step process:

  1. Use the product rule, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

  2. Let ( u = x^3 - 8 ) and ( v = \tan^2(5x - 3) ).

  3. Find the derivatives of ( u ) and ( v ): ( u' = 3x^2 ) ( v' = 2\tan(5x - 3)\sec^2(5x - 3)\cdot5 )

  4. Apply the product rule: ( f'(x) = u'v + uv' ) ( f'(x) = (3x^2)\tan^2(5x - 3) + (x^3 - 8)2\tan(5x - 3)\sec^2(5x - 3)\cdot5 )

  5. Simplify the expression, if necessary.

So, the derivative of ( f(x) = (x^3 - 8)\tan^2(5x - 3) ) is ( f'(x) = 3x^2\tan^2(5x - 3) + 10(x^3 - 8)\tan(5x - 3)\sec^2(5x - 3) ).

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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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