I suppose you mean
![g(x) = \dfrac x{2\sqrt{36-x^2}} + 18\sin^{-1}\left(\dfrac x6\right)](https://tex.z-dn.net/?f=g%28x%29%20%3D%20%5Cdfrac%20x%7B2%5Csqrt%7B36-x%5E2%7D%7D%20%2B%2018%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%20x6%5Cright%29)
Differentiate one term at a time.
Rewrite the first term as
![\dfrac x{2\sqrt{36-x^2}} = \dfrac12 x(36-x^2)^{-1/2}](https://tex.z-dn.net/?f=%5Cdfrac%20x%7B2%5Csqrt%7B36-x%5E2%7D%7D%20%3D%20%5Cdfrac12%20x%2836-x%5E2%29%5E%7B-1%2F2%7D)
Then the product rule says
![\left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 x' (36-x^2)^{-1/2} + \dfrac12 x \left((36-x^2)^{-1/2}\right)'](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac12%20x%2836-x%5E2%29%5E%7B-1%2F2%7D%5Cright%29%27%20%3D%20%5Cdfrac12%20x%27%20%2836-x%5E2%29%5E%7B-1%2F2%7D%20%2B%20%5Cdfrac12%20x%20%5Cleft%28%2836-x%5E2%29%5E%7B-1%2F2%7D%5Cright%29%27)
Then with the power and chain rules,
![\left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-1/2} + \dfrac12\left(-\dfrac12\right) x (36-x^2)^{-3/2}(36-x^2)' \\\\ \left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-1/2} - \dfrac14 x (36-x^2)^{-3/2} (-2x) \\\\ \left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-1/2} + \dfrac12 x^2 (36-x^2)^{-3/2}](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac12%20x%2836-x%5E2%29%5E%7B-1%2F2%7D%5Cright%29%27%20%3D%20%5Cdfrac12%20%2836-x%5E2%29%5E%7B-1%2F2%7D%20%2B%20%5Cdfrac12%5Cleft%28-%5Cdfrac12%5Cright%29%20x%20%2836-x%5E2%29%5E%7B-3%2F2%7D%2836-x%5E2%29%27%20%5C%5C%5C%5C%20%5Cleft%28%5Cdfrac12%20x%2836-x%5E2%29%5E%7B-1%2F2%7D%5Cright%29%27%20%3D%20%5Cdfrac12%20%2836-x%5E2%29%5E%7B-1%2F2%7D%20-%20%5Cdfrac14%20x%20%2836-x%5E2%29%5E%7B-3%2F2%7D%20%28-2x%29%20%5C%5C%5C%5C%20%5Cleft%28%5Cdfrac12%20x%2836-x%5E2%29%5E%7B-1%2F2%7D%5Cright%29%27%20%3D%20%5Cdfrac12%20%2836-x%5E2%29%5E%7B-1%2F2%7D%20%2B%20%5Cdfrac12%20x%5E2%20%2836-x%5E2%29%5E%7B-3%2F2%7D)
Simplify this a bit by factoring out
:
![\left(\dfrac12 x(36-x^2)^{-1/2}\right)' = \dfrac12 (36-x^2)^{-3/2} \left((36-x^2) + x^2\right) = 18 (36-x^2)^{-3/2}](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac12%20x%2836-x%5E2%29%5E%7B-1%2F2%7D%5Cright%29%27%20%3D%20%5Cdfrac12%20%2836-x%5E2%29%5E%7B-3%2F2%7D%20%5Cleft%28%2836-x%5E2%29%20%2B%20x%5E2%5Cright%29%20%3D%2018%20%2836-x%5E2%29%5E%7B-3%2F2%7D)
For the second term, recall that
![\left(\sin^{-1}(x)\right)' = \dfrac1{\sqrt{1-x^2}}](https://tex.z-dn.net/?f=%5Cleft%28%5Csin%5E%7B-1%7D%28x%29%5Cright%29%27%20%3D%20%5Cdfrac1%7B%5Csqrt%7B1-x%5E2%7D%7D)
Then by the chain rule,
![\left(18\sin^{-1}\left(\dfrac x6\right)\right)' = 18 \left(\sin^{-1}\left(\dfrac x6\right)\right)' \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{18\left(\frac x6\right)'}{\sqrt{1 - \left(\frac x6\right)^2}} \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{18\left(\frac16\right)}{\sqrt{1 - \frac{x^2}{36}}} \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{3}{\frac16\sqrt{36 - x^2}} \\\\ \left(18\sin^{-1}\left(\dfrac x6\right)\right)' = \dfrac{18}{\sqrt{36 - x^2}} = 18 (36-x^2)^{-1/2}](https://tex.z-dn.net/?f=%5Cleft%2818%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%20x6%5Cright%29%5Cright%29%27%20%3D%2018%20%5Cleft%28%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%20x6%5Cright%29%5Cright%29%27%20%5C%5C%5C%5C%20%5Cleft%2818%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%20x6%5Cright%29%5Cright%29%27%20%3D%20%5Cdfrac%7B18%5Cleft%28%5Cfrac%20x6%5Cright%29%27%7D%7B%5Csqrt%7B1%20-%20%5Cleft%28%5Cfrac%20x6%5Cright%29%5E2%7D%7D%20%5C%5C%5C%5C%20%5Cleft%2818%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%20x6%5Cright%29%5Cright%29%27%20%3D%20%5Cdfrac%7B18%5Cleft%28%5Cfrac16%5Cright%29%7D%7B%5Csqrt%7B1%20-%20%5Cfrac%7Bx%5E2%7D%7B36%7D%7D%7D%20%5C%5C%5C%5C%20%5Cleft%2818%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%20x6%5Cright%29%5Cright%29%27%20%3D%20%5Cdfrac%7B3%7D%7B%5Cfrac16%5Csqrt%7B36%20-%20x%5E2%7D%7D%20%5C%5C%5C%5C%20%5Cleft%2818%5Csin%5E%7B-1%7D%5Cleft%28%5Cdfrac%20x6%5Cright%29%5Cright%29%27%20%3D%20%5Cdfrac%7B18%7D%7B%5Csqrt%7B36%20-%20x%5E2%7D%7D%20%3D%2018%20%2836-x%5E2%29%5E%7B-1%2F2%7D)
So we have
![g'(x) = 18 (36-x^2)^{-3/2} + 18 (36-x^2)^{-1/2}](https://tex.z-dn.net/?f=g%27%28x%29%20%3D%2018%20%2836-x%5E2%29%5E%7B-3%2F2%7D%20%2B%2018%20%2836-x%5E2%29%5E%7B-1%2F2%7D)
and we can simplify this by factoring out
to end up with
![g'(x) = 18(36-x^2)^{-3/2} \left(1 + (36-x^2)\right) = \boxed{18 (36 - x^2)^{-3/2} (37-x^2)}](https://tex.z-dn.net/?f=g%27%28x%29%20%3D%2018%2836-x%5E2%29%5E%7B-3%2F2%7D%20%5Cleft%281%20%2B%20%2836-x%5E2%29%5Cright%29%20%3D%20%5Cboxed%7B18%20%2836%20-%20x%5E2%29%5E%7B-3%2F2%7D%20%2837-x%5E2%29%7D)
Answer:
A card balance is the total amount of money that you currently owe on your credit card. The balance increases when purchases are made and decreases when payments are made. Purchases, balance transfers, foreign exchange, fees, and interest all factor into your credit card balance.
please brainlest and like
Different examples above! hope this helps c:
75.89 + 0.12x = 27.41 + 0.36x
75.89 - 27.41 = 0.24x
48.48 = 0.24x
x = 48.48 / 0.24
x = 202
Each sent 202 text messages