91Ó°ÊÓ

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ \frac{d F}{d u}\left(\frac{\pi}{6}\right), F(u)=5 u+6 \cos (u) $$

\( y\) is a function of \(x .\) Calculate the derivative of the given expression with respect to \(x\). (Your answer should contain the term \(d y / d x .)\) \(2 x / \sqrt{y}\)

Assume that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is invertible and differentiable. Compute \(\left(f^{-1}\right)^{\prime}(4)\) from the given information. $$ f^{-1}(4)=-1, f^{\prime}(-1)=3 $$

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(1 / x\)

Calculate the derivative of the given expression with respect to \(x\). $$ e^{5 x} $$

Calculate the value of the given inverse trigonometric function at the given point. $$ \arccos (-1) $$

Calculate the derivative of the given expression with respect to \(x\). $$ \sec (x / 4) $$

Use the rules for differentiating sums and differences, as in Example \(1,\) to compute the derivative of the given expression with respect to \(x\) $$ 3 x^{3}-2 x^{2}+\pi \sin (x)+1 / \pi $$

\( y\) is a function of \(x .\) Calculate the derivative of the given expression with respect to \(x\). (Your answer should contain the term \(d y / d x .)\) \(\left(y^{3}-1\right) / y\)

Assume that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is invertible and differentiable. Compute \(\left(f^{-1}\right)^{\prime}(4)\) from the given information. $$ f^{-1}(4)=7, f^{\prime}(7)=-3 / 8 $$