What the slope of #x^4y^4=16# at #(2,1) #?

Answer 1

The slope (of the tangent) to the curve at the given coordinate is
# -1/2 #

We seek the slope (of the tangent) to the curve:

# x^4y^4 = 16 #
at the coordinate #(2,1)#.

The gradient of the tangent to a curve at any particular point is given by the derivative of the curve at that point. So if we differentiate the equation implicitly, and apply the product rule, then we have:

# (x^4)(d/dx y^4) + (d/dx x^4)(y^4) = 0 #
# :. (x^4)(4y^3 dy/dx) + (4x^3)(y^4) = 0 #
So if the slope of the tangent at the given coordinate #(2,1)# is #m#, then we have:
# :. (2^4)(4*1^3 *m) + (4*2^3)(1^4) = 0 #
# :. 16(4m) + 4*8 = 0 #
# :. 2m + 1 = 0 #
# :. m =-1/2 #
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Answer 2

To find the slope of the curve at a specific point, we need to differentiate the equation implicitly with respect to x and y.

Differentiating both sides of the equation x^4y^4 = 16 with respect to x, we get: 4x^3y^4 + 4y^3x^4(dy/dx) = 0

To find the slope at the point (2,1), we substitute x = 2 and y = 1 into the equation: 4(2)^3(1)^4 + 4(1)^3(2)^4(dy/dx) = 0

Simplifying this equation, we have: 32 + 128(dy/dx) = 0

Solving for dy/dx, we get: dy/dx = -32/128 = -1/4

Therefore, the slope of the curve x^4y^4 = 16 at the point (2,1) is -1/4.

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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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