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![\huge \boxed{\mathfrak{Question} \downarrow}](https://tex.z-dn.net/?f=%20%5Chuge%20%5Cboxed%7B%5Cmathfrak%7BQuestion%7D%20%5Cdownarrow%7D)
![\huge \tt\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }](https://tex.z-dn.net/?f=%20%5Chuge%20%5Ctt%5Cfrac%20%7B%20d%20%7D%20%7B%20d%20x%20%7D%20%5Cfrac%20%7B%20%28%203%20x%20%5E%20%7B%202%20%7D%20-%202%20%29%20%7D%20%7B%20%28%20x%20-%205%20%29%20%7D)
![\large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}](https://tex.z-dn.net/?f=%20%5Clarge%20%5Cboxed%7B%5Cmathfrak%7BAnswer%20%5C%3A%20with%20%5C%3A%20Explanation%7D%20%5Cdownarrow%7D)
![\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) } \\](https://tex.z-dn.net/?f=%5Cfrac%20%7B%20d%20%7D%20%7B%20d%20x%20%7D%20%5Cfrac%20%7B%20%28%203%20x%20%5E%20%7B%202%20%7D%20-%202%20%29%20%7D%20%7B%20%28%20x%20-%205%20%29%20%7D%20%5C%5C%20)
To start solving this question, note that ⇨ for any 2 differentiable functions, the derivative of the quotient of the 2 functions will be the denominator multiplied by the derivative of the numerator minus the numerator again multiplied by the derivative of the denominator whole divided by the denominator². By doing all these steps, we'll get it as..
![\frac{\left(x^{1}-5\right)\frac{\mathrm{d}}{\mathrm{d}x}(3x^{2}-2)-\left(3x^{2}-2\right)\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}-5)}{\left(x^{1}-5\right)^{2}} \\](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5Cfrac%7B%5Cmathrm%7Bd%7D%7D%7B%5Cmathrm%7Bd%7Dx%7D%283x%5E%7B2%7D-2%29-%5Cleft%283x%5E%7B2%7D-2%5Cright%29%5Cfrac%7B%5Cmathrm%7Bd%7D%7D%7B%5Cmathrm%7Bd%7Dx%7D%28x%5E%7B1%7D-5%29%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20)
Remember that the derivative of the polynomial will be the sum of the derivatives of its terms. We know that, the derivative of a constant term is 0 & the derivative of
is
. So..
![\frac{\left(x^{1}-5\right)\times 2\times 3x^{2-1}-\left(3x^{2}-2\right)x^{1-1}}{\left(x^{1}-5\right)^{2}} \\](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5Ctimes%202%5Ctimes%203x%5E%7B2-1%7D-%5Cleft%283x%5E%7B2%7D-2%5Cright%29x%5E%7B1-1%7D%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20)
Now, simplify it..
![\frac{\left(x^{1}-5\right)\times 2\times 3x^{2-1}-\left(3x^{2}-2\right)x^{1-1}}{\left(x^{1}-5\right)^{2}} \\ = \frac{\left(x^{1}-5\right)\times 6x^{1}-\left(3x^{2}-2\right)x^{0}}{\left(x^{1}-5\right)^{2}} \\ = \frac{x^{1}\times 6x^{1}-5\times 6x^{1}-\left(3x^{2}x^{0}-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{6x^{1+1}-5\times 6x^{1}-\left(3x^{2}-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{6x^{2}-30x^{1}-\left(3x^{2}-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{6x^{2}-30x^{1}-3x^{2}-\left(-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{\left(6-3\right)x^{2}-30x^{1}-\left(-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{3x^{2}-30x^{1}-\left(-2x^{0}\right)}{\left(x^{1}-5\right)^{2}} \\ = \frac{3x^{2}-30x-\left(-2\right)}{\left(x-5\right)^{2}} \\ = \large \boxed{\boxed{ \bf \frac{3x^{2}-30x + 2}{\left(x-5\right)^{2}} }}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5Ctimes%202%5Ctimes%203x%5E%7B2-1%7D-%5Cleft%283x%5E%7B2%7D-2%5Cright%29x%5E%7B1-1%7D%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%5Cfrac%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5Ctimes%206x%5E%7B1%7D-%5Cleft%283x%5E%7B2%7D-2%5Cright%29x%5E%7B0%7D%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%5Cfrac%7Bx%5E%7B1%7D%5Ctimes%206x%5E%7B1%7D-5%5Ctimes%206x%5E%7B1%7D-%5Cleft%283x%5E%7B2%7Dx%5E%7B0%7D-2x%5E%7B0%7D%5Cright%29%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%5Cfrac%7B6x%5E%7B1%2B1%7D-5%5Ctimes%206x%5E%7B1%7D-%5Cleft%283x%5E%7B2%7D-2x%5E%7B0%7D%5Cright%29%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%20%5C%5C%20%20%3D%20%5Cfrac%7B6x%5E%7B2%7D-30x%5E%7B1%7D-%5Cleft%283x%5E%7B2%7D-2x%5E%7B0%7D%5Cright%29%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%5Cfrac%7B6x%5E%7B2%7D-30x%5E%7B1%7D-3x%5E%7B2%7D-%5Cleft%28-2x%5E%7B0%7D%5Cright%29%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%5Cfrac%7B%5Cleft%286-3%5Cright%29x%5E%7B2%7D-30x%5E%7B1%7D-%5Cleft%28-2x%5E%7B0%7D%5Cright%29%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%5Cfrac%7B3x%5E%7B2%7D-30x%5E%7B1%7D-%5Cleft%28-2x%5E%7B0%7D%5Cright%29%7D%7B%5Cleft%28x%5E%7B1%7D-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%5Cfrac%7B3x%5E%7B2%7D-30x-%5Cleft%28-2%5Cright%29%7D%7B%5Cleft%28x-5%5Cright%29%5E%7B2%7D%7D%20%20%5C%5C%20%20%3D%20%20%5Clarge%20%5Cboxed%7B%5Cboxed%7B%20%5Cbf%20%5Cfrac%7B3x%5E%7B2%7D-30x%20%2B%202%7D%7B%5Cleft%28x-5%5Cright%29%5E%7B2%7D%7D%20%7D%7D)
- You can further simplify the answer to
![\underline{\underline{\frac{146}{\left(x-5\right)^{3}}}}\\](https://tex.z-dn.net/?f=%5Cunderline%7B%5Cunderline%7B%5Cfrac%7B146%7D%7B%5Cleft%28x-5%5Cright%29%5E%7B3%7D%7D%7D%7D%5C%5C)
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Hope it'll help you!
ℓu¢αzz ッ
Answer:
Step-by-step explanation:
Yes it 120
Answer:
The slope is around -2.5
Step-by-step explanation:
pls dont hurt me if im wrong i cant quite see the full line and the y intercept is not on the graph