Chapter 3

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

In each of Exercises \(1-6, y\) is a function of \(x .\) Calculate the derivative of the given expression with respect to \(x\). (Your answer should contain the term \(d y / d x .)\) \(4 y^{3 / 2}\)

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(5 x^{2}-6 x+7\)

Calculate the derivative of the given expression with respect to \(x\). $$ \sin (3 x) $$

In Exercises \(1-8,\) assume that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is invertible and differentiable. Compute \(\left(f^{-1}\right)^{\prime}(4)\) from the given information. $$ f^{-1}(4)=1, f^{\prime}(1)=2 $$

In Exercises \(1-28\) differentiate the given expression with respect to \(x\). \(8 x^{10}-6 x^{-5}\)

In Exercises 1-6, use the rules for differentiating sums and differences, as in Example \(1,\) to compute the derivative of the given expression with respect to \(x\) $$ 5 x^{3}-3 x^{2}+9 $$

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ f^{\prime}(-3), f(x)=\sin (x)-\cos (x) $$

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

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=\sqrt{x}, c=9, x=8.95 $$

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

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(4 x^{3}-7 x^{-5}\)

Assume that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is invertible and differentiable. Compute \(\left(f^{-1}\right)^{\prime}(4)\) from the given information. $$ f^{-1}(4)=3, f^{\prime}(3)=1 / 6 $$

Differentiate the given expression with respect to \(x\). \(2 x^{3 / 2}-1 / x^{3 / 2}\)

Use the rules for differentiating sums and differences, as in Example \(1,\) to compute the derivative of the given expression with respect to \(x\) $$ 6-4 / x+4 x-x^{2} / 2-2 x^{3} $$

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ \dot{g}(\pi / 2), g(t)=t^{3} / 3+\cos (t) $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=1 / x^{1 / 3}, c=8, x=8.07 $$

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

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

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

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

Differentiate the given expression with respect to \(x\). \(6 x^{5 / 3}-25 x^{3 / 5}\)

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

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

Describes the position of an object at time \(t .\) Calculate the instantaneous velocity at time \(c\). \(p(t)=-7 t^{2} \quad c=3\)

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

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=x^{-3 / 2}, c=4, x=4.21 $$

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

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(x^{7 / 3}\)

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

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

Differentiate the given expression with respect to \(x\). \(\sqrt{x^{5}}+3 / \sqrt{x}\)

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

Describes the position of an object at time \(t .\) Calculate the instantaneous velocity at time \(c\). \(p(t)=-4 t^{3} \quad c=2\)

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ D(f)(0), f(x)=-3 \sin (x) $$

Calculate the value of the given inverse trigonometric function at the given point. $$ \arccos (1 / 2) $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=x^{2 / 3}, c=8, x=8.15 $$

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

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

Calculate the derivative of the given expression with respect to \(x\). $$ \cos (1 / x) $$

Differentiate the given expression with respect to \(x\). \(2 x+e^{x}\)

Use the rules for differentiating sums and differences, as in Example \(1,\) to compute the derivative of the given expression with respect to \(x\) $$ \frac{1}{5}(4 \sin (x)-3 \cos (x))+5 x $$

Describes the position of an object at time \(t .\) Calculate the instantaneous velocity at time \(c\). $$ p(t)=-4 t^{3} \quad c=2 $$

Compute the indicated derivative for the given function by using the formulas and rules that are summarized at the end of this section. $$ \dot{g}(5), g(t)=8 t^{3}-8 t $$

Calculate the value of the given inverse trigonometric function at the given point. $$ \arccos (-\sqrt{3} / 2) $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=(1+x)^{-1 / 4}, c=15, x=16 $$

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

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

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

Calculate the derivative of the given expression with respect to \(x\). $$ \sin (\sqrt{x}) $$