Limits, Continuity and Differentiability
80322
If \(x=f(t)\) and \(y=g(t)\) are differentiable functions of \(t\), then \(\frac{d^{2} y}{d x^{2}}\) is
1 \(\frac{f^{\prime}(t) \cdot g^{\prime \prime}(t)-g^{\prime}(t) \cdot f^{\prime \prime}(t)}{\left[f^{\prime}(t)\right]^{3}}\)
2 \(\frac{f^{\prime}(t) \cdot g^{\prime \prime}(t)-g^{\prime}(t) \cdot f^{\prime \prime}(t)}{\left[f^{\prime}(t)\right]^{2}}\)
3 \(\frac{g^{\prime}(t) \cdot f^{\prime \prime}(t)-f^{\prime}(t) \cdot g^{\prime \prime}(t)}{\left[f^{\prime}(t)\right]^{3}}\)
4 \(\frac{g^{\prime}(g) \cdot f^{\prime \prime}(t)+f^{\prime}(t) \cdot g^{\prime \prime}(t)}{\left[f^{\prime}(t)\right]^{3}}\)
Explanation:
(A) : Given, \(\mathrm{x}=f(\mathrm{t})\)
\(\frac{\mathrm{dx}}{\mathrm{dt}}=f^{\prime}(\mathrm{t})\)
And, \(\mathrm{y}=\mathrm{g}(\mathrm{t})\)
\(\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{g}^{\prime}(\mathrm{t})\)
Now, \(\frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x} \Rightarrow \frac{d y}{d x}=\frac{d y / d t}{d x / d t} \Rightarrow \frac{d y}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)}\)
\(\therefore \quad \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\frac{\mathrm{d}}{\mathrm{dt}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}{\mathrm{dx} / \mathrm{dt}}\)
\(\frac{d^{2} y}{d x^{2}}=\frac{f^{\prime}(t) \cdot g^{\prime \prime}(t)-g^{\prime}(t) \cdot f^{\prime \prime}(t)}{\left(f^{\prime}(t)\right)^{2}}\)
\(\frac{d^{2} y}{d x^{2}}=\frac{f^{\prime}(t) \cdot g^{\prime \prime}(t)-g^{\prime}(t) \cdot f^{\prime \prime}(t)}{\left[f^{\prime}(t)\right]^{3}}\)