For the function \( f(x)=5 x+1 \), find \( (a) f(x+h),(b) f(x+h)-f(x) \), and \( (c) \frac{f(x+h)-f(x)}{h} \)

\( \lim _ { x \rightarrow \infty } ( \sqrt { 25 x ^ { 2 } + 20 x } - \sqrt { x ^ { 2 } - 2 x } - \sqrt { 16 x ^ { 2 } + 16 x } ) \)

3)- Pour tout \( n \in \mathbb{N}^{*} \), On pose : \( S_{n}=\sum_{k=n+1}^{2 n} \frac{g^{-1}\left(u_{k}\right)}{k} \) a) - Montrer que : \( \left(\forall n \in \mathbb{N}^{*}\right), g^{-1}\left(u_{n+1}\right) \cdot \sum_{k=n+1}^{2 n} \frac{1}{k} \leq S_{n} \leq g^{-1}\left(u_{2 n}\right) \cdot \sum_{k=n+1}^{2 n} \frac{1}{k} \) b)- Déduire la limite de la suite \( \left(S_{n}\right)_{n \in \mathbb{N}^{*}} \) \( \quad\left(\right. \) On admet que \( \left.\lim _{n \rightarrow+\infty} \sum_{k=n+1}^{2 n} \frac{1}{k}=\ln 2\right) \)

Find the arclength of \( y=4 x-5 \) on \( 0 \leq x \leq 3 \)

Find the derivative of \( \begin{array}{l}y=4 e^{\sin x} \\ y=2 x^{5} \cos x\end{array} \)

Find the arclength of \( y=4 x^{3 / 2} \) on \( 2 \leq x \leq 4 \)