My first one is express d y d x in terms of x and y if x 2 − 4 x y 3 + 8 x 2 y = 20what I did is this 2 x − 4 x ( 3 y 2 ) ( d y d x ) + 4 y 3 + 8 x 2 d y d x + 16 x y = 0 − 4 x ( 3 y 2 d y d x ) + 8 x 2 d y d x = − 2 x − 4 y 3 − 16Finally I got d y d x = 2 x − 4 y 3 − 16 − 4 ( x ) ( 3 y 2 ) + 8 x 2 My second question isFind an equation for the tangent line to the graph of 2 x 2 − 5 y 2 + x y = 5 at the point (2,1)Anyway I simplified the equation to d y d x = − 4 x − 1 − 10 y + x plugging in x and y I got 9 8 so my line is y − 1 = 9 8 ( x − 2 ) but I am unsure if if I did this correctly.

Question

My first one is express             d      y              d      x       in terms of   x and   y if       x    2    −  4  x      y    3    +  8      x    2    y  =  20what I did is this  2  x  −  4  x  (  3      y    2    )  (            d      y              d      x        )  +  4      y    3    +  8      x    2              d      y              d      x        +  16  x  y  =  0  −  4  x  (  3      y    2              d      y              d      x        )  +  8      x    2              d      y              d      x        =  −  2  x  −  4      y    3    −  16Finally I got             d      y              d      x        =            2      x      −      4              y        3            −      16              −      4      (      x      )      (      3              y        2            )      +      8              x        2            My second question isFind an equation for the tangent line to the graph of   2      x    2    −  5      y    2    +  x  y  =  5 at the point (2,1)Anyway I simplified the equation to            d      y              d      x        =            −      4      x      −      1              −      10      y      +      x       plugging in x and y I got       9    8  so my line is   y  −  1  =      9    8    (  x  −  2  ) but I am unsure if if I did this correctly.

rae2721 · Accepted Answer

Very well done. You indeed found the equation for the line tangent to the second equation at the point (2,1), and your first solution look like you differentiated properly, too.You could simplify just a tad:            d      y              d      x        =            2      x      −      4              y        3            −      16              −      4      (      x      )      (      3              y        2            )      +      8              x        2              =            2      (      x      −      2              y        3            −      8      )              4      (      −      3      x              y        2            +      2      )        =            x      −      2              y        3            −      8              2      (      −      3      x              y        2            +      2      )

HypeMyday3m · Accepted Answer

There are slight computation errors and omission errors; Other than that you are doing fine. If you experience such errors, it is better to write every step.In the first one, when you differentiate both sides you get   2  x  −  4  (  x  3      y    2    d  y      /    d  x  +      y    3    )  +  8  (  2  x  y  +      x    2    d  y      /    d  x  )  =  0. Thus   (  −  4  x  3      y    2    +  8      x    2    )  d  y      /    d  x  =  −  2  x  +  4      y    3    −  16  x  y so that   d  y      /    d  x  =            −      2      x      +      4              y        3            −      16      x      y              −      12      x              y        2            +      8              x        2              =            −      x      +      2              y        3            −      8      x      y              −      6      x              y        2            +      4              x        2            In the second one, a similar error is there. Differentiating gives   4  x  −  10  y  d  y      /    d  x  +  (  x  d  y      /    d  x  +  y  )  =  0 and thus   (  −  10  y  +  x  )  d  y      /    d  x  =  −  4  x  +  y so that   d  y      /    d  x  =            −      4      x      −      y              −      10      y      +      x      . The value you get after substitution is accidentally(?) same here.

My first one is express (dy)/(dx) in terms of x and y if x^2−4xy63+8x^2y=20 My second question is Find an equation for the tangent line to the graph of 2x^2−5y^2+xy=5 at the point (2,1)

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New Questions in Calculus 2