How do you differentiate implicitly to find the slope of the curve #y^4 + x^3 = y^2 + 10x# at the given point (0,1)?

Answer 1
#y^4 + x^3 = y^2 + 10x#
Differentiate both sides of the equation with respect to #x#.

Use whichever notation you prefer:

#D_x(y^4 + x^3) = D_x(y^2 + 10x)#
#d/dx(y^4 + x^3) =d/dx( y^2 + 10x)#
#4y^3 dy/dx +3x^2 = 2y dy/dx +10#
Now, if we want the general formula for #dy/dx#, solve algebraically for #dy/dx#, but all we have been aksed for is #dy/dx# when #x=0# and #y=1#, so let's just do that:
#4(1)^3 dy/dx +3(0)^2 = 2(1) dy/dx +10#
#4 dy/dx = 2 dy/dx + 10#
#dy/dx = 10/2 =5#
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Answer 2

To find the slope of the curve ( y^4 + x^3 = y^2 + 10x ) implicitly at the point ( (0,1) ), differentiate both sides of the equation with respect to ( x ), then solve for ( \frac{{dy}}{{dx}} ).

Differentiating both sides of the equation: [ \frac{{d}}{{dx}}(y^4) + \frac{{d}}{{dx}}(x^3) = \frac{{d}}{{dx}}(y^2) + \frac{{d}}{{dx}}(10x) ]

[ 4y^3\frac{{dy}}{{dx}} + 3x^2 = 2y\frac{{dy}}{{dx}} + 10 ]

Now, plug in the point ( (0,1) ) into the equation and solve for ( \frac{{dy}}{{dx}} ): [ 4(1)^3\frac{{dy}}{{dx}} + 3(0)^2 = 2(1)\frac{{dy}}{{dx}} + 10 ]

[ 4\frac{{dy}}{{dx}} = 2\frac{{dy}}{{dx}} + 10 ]

[ 2\frac{{dy}}{{dx}} = 10 ]

[ \frac{{dy}}{{dx}} = 5 ]

Therefore, the slope of the curve at the point ( (0,1) ) is ( 5 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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