How do you find f" given #f(x)= (6x + 5)^(1/3)#?

Answer 1

# f''(x)=-8/(6x+5)^(5/3)#.

#f(x)=(6x+5)^(1/3)#
#rArr f'(x)=1/3*(6x+5)^(1/3-1)*d/dx(6x+5)............["Chain Rule"]#
#=6/(3(6x+5)^(2/3))#
# f'(x)=2/(6x+5)^(2/3)#
Now, #f''(x)={f'(x)}'#
#:. f''(x)=2(-2/3)(6x+5)^(-2/3-1)*6#
#:. f''(x)=-8/(6x+5)^(5/3)#.
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Answer 2

To find the second derivative ( f''(x) ) of the function ( f(x) = (6x + 5)^{\frac{1}{3}} ), follow these steps:

  1. Find the first derivative ( f'(x) ) using the power rule and chain rule.
  2. Once you have ( f'(x) ), differentiate it again to find ( f''(x) ).

Step 1: Finding the first derivative ( f'(x) ): [ f(x) = (6x + 5)^{\frac{1}{3}} ] [ f'(x) = \frac{1}{3}(6x + 5)^{-\frac{2}{3}} \cdot 6 ] [ f'(x) = 2(6x + 5)^{-\frac{2}{3}} ]

Step 2: Finding the second derivative ( f''(x) ): Using the power rule and chain rule again: [ f''(x) = \frac{d}{dx}[2(6x + 5)^{-\frac{2}{3}}] ] [ f''(x) = 2 \cdot (-\frac{2}{3})(6x + 5)^{-\frac{2}{3}-1} \cdot 6 ] [ f''(x) = -8(6x + 5)^{-\frac{5}{3}} ]

Therefore, ( f''(x) = -8(6x + 5)^{-\frac{5}{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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