Answer:
Step-by-step explanation:
The angle of 2 is 75 degrees.
Angle 1 = 180 - 75 = 105 degrees
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Find the derivative of
![\mathsf{y=3x\cdot sin^2\big(x^{1/2}\big)}](https://tex.z-dn.net/?f=%5Cmathsf%7By%3D3x%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%7D)
First, differentiate it by applying the product rule:
![\mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}\!\left[3x\cdot sin^2\big(x^{1/2}\big)\right]}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}(3x)\cdot sin^2\big(x^{1/2}\big)+3x\cdot \dfrac{d}{dx}\!\left[sin^2\big(x^{1/2}\big)\right]}\\\\\\ \mathsf{\dfrac{dy}{dx}=3\cdot sin^2\big(x^{1/2}\big)+3x\cdot \dfrac{d}{dx}\!\left[sin^2\big(x^{1/2}\big)\right]}\\\\\\ \mathsf{\dfrac{dy}{dx}=3\cdot sin^2\big(x^{1/2}\big)+3x\cdot \dfrac{du}{dx}\qquad\quad(i)}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D%5Cdfrac%7Bd%7D%7Bdx%7D%5C%21%5Cleft%5B3x%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D%5Cdfrac%7Bd%7D%7Bdx%7D%283x%29%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%2B3x%5Ccdot%20%5Cdfrac%7Bd%7D%7Bdx%7D%5C%21%5Cleft%5Bsin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D3%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%2B3x%5Ccdot%20%5Cdfrac%7Bd%7D%7Bdx%7D%5C%21%5Cleft%5Bsin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D3%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%2B3x%5Ccdot%20%5Cdfrac%7Bdu%7D%7Bdx%7D%5Cqquad%5Cquad%28i%29%7D)
where
![\mathsf{u=sin^2\big(x^{1/2}\big)}\\\\\\\left\{\! \begin{array}{l} \mathsf{u=v^2}\\\\ \mathsf{v=sin\,w}\\\\ \mathsf{w=x^{1/2}} \end{array} \right.](https://tex.z-dn.net/?f=%5Cmathsf%7Bu%3Dsin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%7D%5C%5C%5C%5C%5C%5C%5Cleft%5C%7B%5C%21%20%5Cbegin%7Barray%7D%7Bl%7D%20%5Cmathsf%7Bu%3Dv%5E2%7D%5C%5C%5C%5C%20%5Cmathsf%7Bv%3Dsin%5C%2Cw%7D%5C%5C%5C%5C%20%5Cmathsf%7Bw%3Dx%5E%7B1%2F2%7D%7D%20%5Cend%7Barray%7D%20%5Cright.)
As
u is a composite function, then you have to apply the chain rule to evaluate its derivative:
![\mathsf{\dfrac{du}{dx}=\dfrac{du}{dv}\cdot \dfrac{dv}{dw}\cdot \dfrac{dw}{dx}}\\\\\\ \mathsf{\dfrac{du}{dx}=\dfrac{d}{dv}(v^2)\cdot \dfrac{d}{dw}(sin\,w)\cdot \dfrac{d}{dx}\big(x^{1/2}\big)}\\\\\\ \mathsf{\dfrac{du}{dx}=\diagup\!\!\!\! 2v^{2-1}\cdot cos\,w\cdot \left[\dfrac{1}{\diagup\!\!\!\! 2}\,x^{(1/2)-1} \right]}\\\\\\ \mathsf{\dfrac{du}{dx}=v\cdot cos\,w\cdot x^{-1/2}}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bdu%7D%7Bdx%7D%3D%5Cdfrac%7Bdu%7D%7Bdv%7D%5Ccdot%20%5Cdfrac%7Bdv%7D%7Bdw%7D%5Ccdot%20%5Cdfrac%7Bdw%7D%7Bdx%7D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdfrac%7Bdu%7D%7Bdx%7D%3D%5Cdfrac%7Bd%7D%7Bdv%7D%28v%5E2%29%5Ccdot%20%5Cdfrac%7Bd%7D%7Bdw%7D%28sin%5C%2Cw%29%5Ccdot%20%5Cdfrac%7Bd%7D%7Bdx%7D%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdfrac%7Bdu%7D%7Bdx%7D%3D%5Cdiagup%5C%21%5C%21%5C%21%5C%21%202v%5E%7B2-1%7D%5Ccdot%20cos%5C%2Cw%5Ccdot%20%5Cleft%5B%5Cdfrac%7B1%7D%7B%5Cdiagup%5C%21%5C%21%5C%21%5C%21%202%7D%5C%2Cx%5E%7B%281%2F2%29-1%7D%20%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdfrac%7Bdu%7D%7Bdx%7D%3Dv%5Ccdot%20cos%5C%2Cw%5Ccdot%20x%5E%7B-1%2F2%7D%7D)
Substitute back for
![\mathsf{v=sin\,w:}](https://tex.z-dn.net/?f=%5Cmathsf%7Bv%3Dsin%5C%2Cw%3A%7D)
![\mathsf{\dfrac{du}{dx}=sin\,w\cdot cos\,w\cdot x^{-1/2}}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bdu%7D%7Bdx%7D%3Dsin%5C%2Cw%5Ccdot%20cos%5C%2Cw%5Ccdot%20x%5E%7B-1%2F2%7D%7D)
and then substitute back for
![\mathsf{w=x^{1/2}:}](https://tex.z-dn.net/?f=%5Cmathsf%7Bw%3Dx%5E%7B1%2F2%7D%3A%7D)
![\mathsf{\dfrac{du}{dx}=sin\big(x^{1/2}\big)\cdot cos\big(x^{1/2}\big)\cdot x^{-1/2}\qquad\quad(ii)}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bdu%7D%7Bdx%7D%3Dsin%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Ccdot%20cos%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Ccdot%20x%5E%7B-1%2F2%7D%5Cqquad%5Cquad%28ii%29%7D)
Subsitute
(ii) into
(i) for
du/dx and you finally get
![\mathsf{\dfrac{dy}{dx}=3\cdot sin^2\big(x^{1/2}\big)+3x\cdot \left[sin\big(x^{1/2}\big)\cdot cos\big(x^{1/2}\big)\cdot x^{-1/2}\right]}\\\\\\ \mathsf{\dfrac{dy}{dx}=3\cdot sin^2\big(x^{1/2}\big)+3x^{1-(1/2)}\cdot sin\big(x^{1/2}\big)\cdot cos\big(x^{1/2}\big)}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D3%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%2B3x%5Ccdot%20%5Cleft%5Bsin%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Ccdot%20cos%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Ccdot%20x%5E%7B-1%2F2%7D%5Cright%5D%7D%5C%5C%5C%5C%5C%5C%20%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D3%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%2B3x%5E%7B1-%281%2F2%29%7D%5Ccdot%20sin%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Ccdot%20cos%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%7D)
![\mathsf{\dfrac{dy}{dx}=3\cdot sin^2\big(x^{1/2}\big)+3x^{1/2}\cdot sin\big(x^{1/2}\big)\cdot cos\big(x^{1/2}\big)}\quad\longleftarrow\quad\textsf{this is the answer.}](https://tex.z-dn.net/?f=%5Cmathsf%7B%5Cdfrac%7Bdy%7D%7Bdx%7D%3D3%5Ccdot%20sin%5E2%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%2B3x%5E%7B1%2F2%7D%5Ccdot%20sin%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%5Ccdot%20cos%5Cbig%28x%5E%7B1%2F2%7D%5Cbig%29%7D%5Cquad%5Clongleftarrow%5Cquad%5Ctextsf%7Bthis%20is%20the%20answer.%7D)
I hope this helps. =)
Tags: <em>derivative composite function product polynomial trigonometric trig sine sin power rule product rule chain rule differential integral calculus</em>
Answer: 53 points
Step-by-step explanation:
Let the number of points gotten by Kate be represented by x.
Since we are given the information that Kevin got 52 points which was 54 points less than 2 times that of kates points. This can be formed into an equation as:
(2 × x) - 54 = 52
2x - 54 = 52
2x = 52 + 54
2x = 106
x = 106/2
x = 53
Kate got 53 points
Answer:
What are you trying to find?
Step-by-step explanation: