Using the expression& 2dsinθ = λ , one calculates the values of d...

2d\text\sin\theta = \lambda$<br />$d = \dfrac{\lambda}{\text{2 sin}\theta}$<br />Differentiate on both the sides,<br />$∂ \left(d\right) = \dfrac{\lambda}{2} ∂ \left(\text{cosec}\theta\right)$<br />$∂ \left(d\right) = \dfrac{\lambda}{2}\left(- \text{cosec}\theta\text{cot}\theta\right) ∂ \theta$<br />$∂ \left(d\right) = \dfrac{- \lambda\text\cos\theta}{2{\text\sin}^{2}\theta} ∂ \theta$<br />$\theta =$increases and$\dfrac{\lambda\text\cos\theta}{2{\text\sin}^{2}\theta}$decreases.<br /><u><strong>Alternate solution</strong></u><br />$d = \dfrac{\lambda}{\text{2 sin}\theta}$<br />$ℓnd = ℓn\lambda - ℓn2 - ℓn\text\sin\theta$<br />$\dfrac{\Delta \left(d\right)}{d} = 0 - 0 - \dfrac{1}{\text\sin\theta} \times \text\cos\theta\left(\Delta \text{\theta}\right)$<br />Fractional error, \left|+ \left(d\right)\right| = \left|\text{cot}\theta \Delta \theta\right|$
Absolute error, $\Delta d = \left(d\text{cot}\theta\right) \Delta \theta$
\dfrac{d}{2\text\sin\theta} \times \dfrac{\text\cos\theta}{\text\sin\theta}
\Delta d = \dfrac{\text\cos\theta}{{\text\sin}^{2}\theta}