Chapter 3

Describes the position of an object at time \(t .\) Calculate the instantaneous velocity at time \(c\). $$ p(t)=1 / t \quad c=1 / 2 $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=\cot (x), c=\pi / 3, x=1 $$

Use the method of implicit differentiation to calculate \(d y / d x\) at the point \(P_{0}\) \(x^{2} y+\ln (y)=1 \quad P_{0}=(-1,1)\)

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(\exp \left(x^{2}\right)\)

Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(0, \infty) \rightarrow(4, \infty), f(s)=s^{2}+4 $$

Calculate the derivative of the given expression with respect to \(x\). $$ \tan \left(x^{3}\right) $$

Differentiate the given expression with respect to \(x\). \(\sec (x)-\tan (x)\)

Calculate \(g^{\prime}(x)\) by using the formulas and rules that are summarized at the end of this section. $$ g(x)=x^{2}-4 x $$

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

Use the method of implicit differentiation to calculate \(d y / d x\) at the point \(P_{0}\) \(x e^{y-1}+y^{2}=4 \quad P_{0}=(3,1)\)

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

Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(0,3) \rightarrow(2,245), f(s)=s^{5}+2 $$

Calculate the derivative of the given expression with respect to \(x\). $$ x / \sqrt{1+x} $$

Differentiate the given expression with respect to \(x\). \(\csc (x)+\cot (x)\)

Use the Product Rule to compute the derivative of the given expression with respect to \(x\). (In each of Exercises 7,8,14,16, and 18, do not avoid using the Product Rule by first expanding the product.) $$ \sin (x) \cos (x) $$

Describes the position of a moving body at time \(t\). Determine whether, at time \(t=4,\) the body is moving forward, backward, or neither. $$ p(t)=6 t+3 $$

Calculate \(g^{\prime}(x)\) by using the formulas and rules that are summarized at the end of this section. $$ g(x)=4 x^{3}+6 x^{2}+1 $$

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

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=\csc (\pi x / 4), c=1, x=0.94 $$

Use the method of implicit differentiation to calculate \(d y / d x\) at the point \(P_{0}\) \(5 x^{3} \frac{x+2 y}{x-y}=0 \quad P_{0}=(1,2)\)

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(\left(x^{2}-1\right)(x+5)\)

Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(-\infty, \infty) \rightarrow(0, \infty), f(s)=\exp (1-s) $$

Calculate the derivative of the given expression with respect to \(x\). $$ x^{2} / \sqrt{1-3 x} $$

Differentiate the given expression with respect to \(x\). \(x \cot (x)-\csc (x)\)

Use the Product Rule to compute the derivative of the given expression with respect to \(x\). (In each of Exercises 7,8,14,16, and 18, do not avoid using the Product Rule by first expanding the product.) $$ \left(x^{2}-5\right)(3 x+2) $$

Describes the position of a moving body at time \(t\). Determine whether, at time \(t=4,\) the body is moving forward, backward, or neither. $$ p(t)=t^{2}-8 t $$

Calculate \(g^{\prime}(x)\) by using the formulas and rules that are summarized at the end of this section. $$ g(x)=2 x+\sin (x) $$

Calculate the value of the given inverse trigonometric function at the given point. $$ \arcsin (\sin (5 \pi / 4)) $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=\ln (x), c=e^{3}, x=20 $$

Use implicit differentiation to find the tangent line to the given curve at the given point \(P_{0}\). \(x y-1=\sqrt{x}+\sqrt{y} \quad P_{0}=(4,1)\)

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

Calculate the derivative of the given expression with respect to \(x\). $$ \cot \left(x^{2}+4\right)+\csc \left(x^{2}+4\right) $$

Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(1,3) \rightarrow(3,15), f(s)=s^{2}+2 s $$

Differentiate the given expression with respect to \(x\). \(\csc (x) \cot (x)\)

Use the Product Rule to compute the derivative of the given expression with respect to \(x\). (In each of Exercises 7,8,14,16, and 18, do not avoid using the Product Rule by first expanding the product.) $$ 7\left(x^{2}+x\right) \sin (x) $$

Describes the position of a moving body at time \(t\). Determine whether, at time \(t=4,\) the body is moving forward, backward, or neither. $$ p(t)=-t^{3}+5 t^{2} $$

Find a point \(x\) where \(f^{\prime}(x)=6\). $$ f(x)=x^{2}+5 $$

Calculate the value of the given inverse trigonometric function at the given point. $$ \arctan (\tan (-3 \pi / 4)) $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=x \ln (x), c=1, x=0.92 $$

Use implicit differentiation to find the tangent line to the given curve at the given point \(P_{0}\). \(x^{3}-y^{3}+4 y=5 \quad P_{0}=(2,-1)\)

An expression for \(f(x)\) is given. Compute the first, second, and third derivatives of \(f(x)\) with respect to \(x\). \(x /(2-x)\)

Calculate the derivative of the given expression with respect to \(x\). $$ \exp \left(x^{2}+1\right) $$

Use the Inverse Function Derivative Rule to calculate \(\left(f^{-1}\right)^{\prime}(t)\). $$ f:(1,8) \rightarrow(1 / 64,1), f(s)=1 / s^{2} $$

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

Use the Product Rule to compute the derivative of the given expression with respect to \(x\). (In each of Exercises 7,8,14,16, and 18, do not avoid using the Product Rule by first expanding the product.) $$ 3 x^{3}\left(x^{3}+7\right) $$

Describes the position of a moving body at time \(t\). Determine whether, at time \(t=4,\) the body is moving forward, backward, or neither. $$ p(t)=1 / t $$

Find a point \(x\) where \(f^{\prime}(x)=6\). $$ f(x)=12 \sin (x) $$

Calculate the value of the given inverse trigonometric function at the given point. $$ \operatorname{arcsec}(\sec (-5 \pi / 6)) $$

Use the method of increments to estimate the value of \(f(x)\) at the given value of \(x\) using the known value \(f(c)\) $$ f(x)=e^{x}, c=0, x=-0.17 $$

Use implicit differentiation to find the tangent line to the given curve at the given point \(P_{0}\). \(\sin ^{2}(x)+\cos ^{2}(y)=5 / 4 \quad P_{0}=(\pi / 3, \pi / 4)\)