Question

The differential of \(f(x)=\log _{e}\left(1+e^{10 x}\right)-\tan ^{-1}\left(e^{5 x}\right)\) at \(x=0\) and for \(d x=0.2\) is
(A) \(0.5\)
(B) \(0.3\)
(C) \(-0.2\)
(D) \(-0.5\)

Answer 1

Ans: (A)
Hint : \(\frac{f(x+d x)-f(x)}{d x}=f^{\prime}(x)\)
$$
f(x+d x)-f(x)=f^{\prime}(x) d x
$$
Now,
$$
f^{\prime}(x)=\frac{10 e^{10 x}}{1+e^{10 x}}-\frac{5 \cdot e^{5 x}}{1+e^{10 x}}=\frac{10 e^{10 x}-5 e^{5 x}}{1+e^{10 x}}
$$
$$
\begin{aligned}
&\left(f^{\prime}(x)\right)_{x=0}=\frac{5}{2} \\
&\Delta f(x)=0.2 \times \frac{5}{2}=0.5
\end{aligned}
$$

Answer 2

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