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)?
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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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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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