91Ó°ÊÓ

Find the first and second derivatives. \begin{equation} y=x^{2}+x+8 \end{equation}

Using the definition, calculate the derivatives of the functions. Then find the values of the derivatives as specified. $$ F(x)=(x-1)^{2}+1 ; \quad F^{\prime}(-1), F^{\prime}(0), F^{\prime}(2) $$

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(d y / d x=f^{\prime}(g(x)) g^{\prime}(x)\) $$ y=2 u^{3}, \quad u=8 x-1 $$

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$s=6 t-t^{2}, \quad 0 \leq t \leq 6$$

Find the linearization \(L(x)\) of \(f(x)\) at \(x=a.\) \(f(x)=\sqrt{x^{2}+9}, \quad a=-4\)

Use implicit differentiation to find \(d y / d x\). \(x^{3}+y^{3}=18 x y\)

Find \(d y / d x\) $$ y=x^{2} \cos x $$

Use implicit differentiation to find \(d y / d x\). \(2 x y+y^{2}=x+y\)

Exercises \(1-6\) give the positions \(s=f(t)\) of a body moving on a coordinate line, with \(s\) in meters and \(t\) in seconds. a. Find the body's displacement and average velocity for the given time interval. b. Find the body's speed and acceleration at the endpoints of the interval. c. When, if ever, during the interval does the body change direction? $$s=-t^{3}+3 t^{2}-3 t, \quad 0 \leq t \leq 3$$

In Exercises \(1-8,\) given \(y=f(u)\) and \(u=g(x),\) find \(d y / d x=\) \(d y / d x=f^{\prime}(g(x)) g^{\prime}(x)\) $$ y=\sin u, \quad u=3 x+1 $$