80194 The differential coefficient of \(\log _{10} x\) with respect to \(\log _{x} 10\) is

80196 If \(y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)\), then \(\frac{d y}{d x}\) is equal to

80197 If \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\), then \(1+\left(\frac{d y}{d x}\right)^{2}\) is _.

80198 If the three functions \(f(x), g(x)\) and \(h(x)\) are such that \(h(x)=f(x) \cdot g(x)\) and \(f^{\prime}(x) \cdot g^{\prime}(x)=c\), where \(c\) is a constant, then \(\frac{\mathrm{f}^{\prime \prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}+\frac{\mathrm{g}^{\prime \prime}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}+\frac{2 \mathrm{c}}{\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})}\) is equal to

80199 If \(f(x)\) is an even functions and \(f^{\prime}(x)\) exists, then \(\mathbf{f}^{\prime}(\mathbf{e})+\mathbf{f}^{\prime}(-\mathbf{e})\) is

80194 The differential coefficient of \(\log _{10} x\) with respect to \(\log _{x} 10\) is

80196 If \(y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)\), then \(\frac{d y}{d x}\) is equal to

80197 If \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\), then \(1+\left(\frac{d y}{d x}\right)^{2}\) is _.

80198 If the three functions \(f(x), g(x)\) and \(h(x)\) are such that \(h(x)=f(x) \cdot g(x)\) and \(f^{\prime}(x) \cdot g^{\prime}(x)=c\), where \(c\) is a constant, then \(\frac{\mathrm{f}^{\prime \prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}+\frac{\mathrm{g}^{\prime \prime}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}+\frac{2 \mathrm{c}}{\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})}\) is equal to

80199 If \(f(x)\) is an even functions and \(f^{\prime}(x)\) exists, then \(\mathbf{f}^{\prime}(\mathbf{e})+\mathbf{f}^{\prime}(-\mathbf{e})\) is

80194 The differential coefficient of \(\log _{10} x\) with respect to \(\log _{x} 10\) is

80196 If \(y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)\), then \(\frac{d y}{d x}\) is equal to

80197 If \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\), then \(1+\left(\frac{d y}{d x}\right)^{2}\) is _.

80198 If the three functions \(f(x), g(x)\) and \(h(x)\) are such that \(h(x)=f(x) \cdot g(x)\) and \(f^{\prime}(x) \cdot g^{\prime}(x)=c\), where \(c\) is a constant, then \(\frac{\mathrm{f}^{\prime \prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}+\frac{\mathrm{g}^{\prime \prime}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}+\frac{2 \mathrm{c}}{\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})}\) is equal to

80199 If \(f(x)\) is an even functions and \(f^{\prime}(x)\) exists, then \(\mathbf{f}^{\prime}(\mathbf{e})+\mathbf{f}^{\prime}(-\mathbf{e})\) is

80194 The differential coefficient of \(\log _{10} x\) with respect to \(\log _{x} 10\) is

80196 If \(y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)\), then \(\frac{d y}{d x}\) is equal to

80197 If \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\), then \(1+\left(\frac{d y}{d x}\right)^{2}\) is _.

80198 If the three functions \(f(x), g(x)\) and \(h(x)\) are such that \(h(x)=f(x) \cdot g(x)\) and \(f^{\prime}(x) \cdot g^{\prime}(x)=c\), where \(c\) is a constant, then \(\frac{\mathrm{f}^{\prime \prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}+\frac{\mathrm{g}^{\prime \prime}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}+\frac{2 \mathrm{c}}{\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})}\) is equal to

80199 If \(f(x)\) is an even functions and \(f^{\prime}(x)\) exists, then \(\mathbf{f}^{\prime}(\mathbf{e})+\mathbf{f}^{\prime}(-\mathbf{e})\) is

80194 The differential coefficient of \(\log _{10} x\) with respect to \(\log _{x} 10\) is

80196 If \(y=\log \left(\frac{1-x^{2}}{1+x^{2}}\right)\), then \(\frac{d y}{d x}\) is equal to

80197 If \(x=a \cos ^{3} \theta, y=a \sin ^{3} \theta\), then \(1+\left(\frac{d y}{d x}\right)^{2}\) is _.

80198 If the three functions \(f(x), g(x)\) and \(h(x)\) are such that \(h(x)=f(x) \cdot g(x)\) and \(f^{\prime}(x) \cdot g^{\prime}(x)=c\), where \(c\) is a constant, then \(\frac{\mathrm{f}^{\prime \prime}(\mathrm{x})}{\mathrm{f}(\mathrm{x})}+\frac{\mathrm{g}^{\prime \prime}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}+\frac{2 \mathrm{c}}{\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})}\) is equal to

80199 If \(f(x)\) is an even functions and \(f^{\prime}(x)\) exists, then \(\mathbf{f}^{\prime}(\mathbf{e})+\mathbf{f}^{\prime}(-\mathbf{e})\) is