Can someone help me with the derivatives and the respective MRS with the questions in the image?

The marginal rate of substitution (MRS) equals the ratio of the marginal utilities with a minus sign.1.

Can’t read your diagram (blurred) but I think this is what is needed...

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1.

MU of X₁ = 4

MU of X₂ = 6

MRS = -4/6 = -2/3 = -0.667

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2.

MU of X₁ = a

MU of X₂ = b

MRS = -a/b

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3.

MU of X₁ = 1/X₁

MU of X₂ = 1

MRS = -(1/X)/1 = -1/X

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4.

MU of X₁ = X₂

MU of X₂ = X₁

MRS = -X₂/X₁

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5

MU of X₁ = aX₁ᵃ⁻¹X₂ᵇ

MU of X₂ = bX₂ᵇ⁻¹X₁ᵃ

MRS = -(aX₁ᵃ⁻¹X₂ᵇ)/(bX₂ᵇ⁻¹X₁ᵃ)

We can simplify this because Xⁿ / Xⁿ⁻¹ = X. E,g. 10⁴/10³ = 10

MRS = -(aX₂)/(bX₁)

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6

MU of X₁ = aX₁ᵃ⁻¹

MU of X₂ = bX₂ᵇ⁻¹

MRS = -(aX₁ᵃ⁻¹)/(bX₂ᵇ⁻¹)