Answer:
>
Step-by-step explanation:
if you change 1/4 into a decimal, it turns into 0.25. So 0.35 is greater than 0.25
Answer:
6x² - 10x + 11
Step-by-step explanation:
Step 1: Write expression
(9x² - 6x) + (5 - 4x) + (6 - 3x²)
Step 2: Combine like terms
6x² - 10x + 11
Answer:
<h2>360 cakes</h2>
Step-by-step explanation:
<h2>soln:</h2><h2>onecake =240\8=30</h2><h2>then 12 cakes =30×12=360</h2>
Answer: 1 1/12
Step-by-step explanation:
1/3 + 3/4 =
1/3 x 4= 4/12
3/4 x 3 = 9/12
9/12 + 4/12 = 13/12 = 1 1/12
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)