Chapter 2

Let \(y=x^{2}+1\) a. Find \(\Delta x\) and \(\Delta y\) if \(x\) changes from 2 to \(2.02\). b. Find the differential \(d y\), and use it to approximate \(\Delta y\) if \(x\) changes from 2 to \(2.02\). c. Compute \(\Delta y-d y\), the error in approximating \(\Delta y\) by \(d y\).

Identify the "inside function" \(u=f(x)\) and the "outside function" \(y=g(u) .\) Then find \(d y / d x\) using the Chain Rule $$ y=(2 x+4)^{3} $$

Find dy/dx by implicit differentiation. $$ 2 x^{2}+y^{2}=4 $$

Differentiate the function. $$ f(x)=\ln (2 x+3) $$

In Exercises, \(s(t)\) is the position function of a body moving along a coordinate line; \(s(t)\) is measured in feet and \(t\) in seconds, where \(t \geq 0 .\) Find the position, velocity, and speed of the body at the indicated time. \(s(t)=1.86 t^{2} ; \quad t=2 \quad\) (free fall on Mars)

Find the derivative of the function. $$ f(x)=4 \cos x-2 x+1 $$

Use the Product Rule to find the derivative of each function. \(f(x)=x^{2} e^{x}\)

Use the definition of the derivative to find the derivative of the function. What is its domain? . \(f(x)=5\)

Let \(y=2 x^{3}-x\). a. Find \(\Delta x\) and \(\Delta y\) if \(x\) changes from 2 to \(1.97\). b. Find the differential \(d y\), and use it to approximate \(\Delta y\) if \(x\) changes from 2 to \(1.97\). c. Compute \(\Delta y-d y\), the error in approximating \(\Delta y\) by \(d y\).

Identify the "inside function" \(u=f(x)\) and the "outside function" \(y=g(u) .\) Then find \(d y / d x\) using the Chain Rule $$ y=\sqrt{x^{2}-4} $$

Find dy/dx by implicit differentiation. $$ y^{2}-3 y=2 x $$

Differentiate the function. $$ g(x)=\ln \left(x^{2}+4\right)^{2} $$

In Exercises, \(s(t)\) is the position function of a body moving along a coordinate line; \(s(t)\) is measured in feet and \(t\) in seconds, where \(t \geq 0 .\) Find the position, velocity, and speed of the body at the indicated time. $$s(t)=2 t^{3}-3 t^{2}+4 t+1 ; \quad t=1$$

Find the derivative of the function. $$ g(x)=x+\tan x $$

Use the Product Rule to find the derivative of each function. \(f(x)=\sqrt{x} e^{x}\)

Find the derivative of the function. \(f(x)=3 x+4\)

Use the definition of the derivative to find the derivative of the function. What is its domain? \(f(x)=2 x+1\)

Let \(w=\sqrt{2 u+3}\). a. Find \(\Delta u\) and \(\Delta w\) if \(u\) changes from 3 to \(3.1\). b. Find the differential \(d w\), and use it to approximate \(\Delta w\) if \(u\) changes from 3 to \(3.1\). c. Compute \(\Delta w-d w\), the error in approximating \(\Delta w\) by \(d w\).

Identify the "inside function" \(u=f(x)\) and the "outside function" \(y=g(u) .\) Then find \(d y / d x\) using the Chain Rule $$ y=\frac{1}{\sqrt[3]{x^{2}+1}} $$

Find dy/dx by implicit differentiation. $$ x y^{2}+y x^{2}-2=0 $$

Differentiate the function. $$ h(x)=\ln \sqrt{x} $$

In Exercises, \(s(t)\) is the position function of a body moving along a coordinate line; \(s(t)\) is measured in feet and \(t\) in seconds, where \(t \geq 0 .\) Find the position, velocity, and speed of the body at the indicated time. $$s(t)=2 t^{4}-8 t^{2}+4 ; \quad t=1$$

Find the derivative of the function. $$ h(t)=3 \tan t-4 \sec t $$

Use the Product Rule to find the derivative of each function. \(f(t)=\sqrt{t}(t+2) e^{t}\)

Find the derivative of the function. \(f(x)=3 x^{2}\)

Use the definition of the derivative to find the derivative of the function. What is its domain? \(f(x)=3 x-4\)

Let \(y=1 / x\). a. Find \(\Delta x\) and \(\Delta y\) if \(x\) changes from 1 to \(1.02\). b. Find the differential \(d y\), and use it to approximate \(\Delta y\) if \(x\) changes from 1 to \(1.02\). c. Compute \(\Delta y-d y\), the error in approximating \(\Delta y\) by \(d y\).

Identify the "inside function" \(u=f(x)\) and the "outside function" \(y=g(u) .\) Then find \(d y / d x\) using the Chain Rule $$ y=2 \sin \pi x $$

Find dy/dx by implicit differentiation. $$ x^{2} y+2 x y^{2}-x+3=0 $$

Differentiate the function. $$ y=\sqrt{\ln x} $$

In Exercises, \(s(t)\) is the position function of a body moving along a coordinate line; \(s(t)\) is measured in feet and \(t\) in seconds, where \(t \geq 0 .\) Find the position, velocity, and speed of the body at the indicated time. $$ s(t)=\frac{t}{t+1} ; \quad t=0 $$

Find the derivative of the function. $$ y=\sqrt{x} \sin x $$

Use the Product Rule to find the derivative of each function. \(f(x)=\frac{e^{x}}{x}\)

Use the definition of the derivative to find the derivative of the function. What is its domain? \(f(x)=2 x^{2}+x\)

Find the differential of the function at the indicated number. \(f(x)=2 x^{2}-3 x+1 ; \quad x=1\)

Identify the "inside function" \(u=f(x)\) and the "outside function" \(y=g(u) .\) Then find \(d y / d x\) using the Chain Rule $$ y=\sqrt{e^{x}+\cos x} $$

Find dy/dx by implicit differentiation. } $$ x^{3}-2 y^{3}-y=x+2 $$

In Exercises \(5-18\), find the differential of the function at the indicated number. $$ f(x)=2 x^{2}-3 x+1 ; \quad x=1 $$

Differentiate the function. $$ g(u)=\ln \frac{u}{u+1} $$

In Exercises, \(s(t)\) is the position function of a body moving along a coordinate line; \(s(t)\) is measured in feet and \(t\) in seconds, where \(t \geq 0 .\) Find the position, velocity, and speed of the body at the indicated time. $$ s(t)=\frac{2 t}{t^{2}+1} ; \quad t=2 $$

Find the derivative of the function. $$ f(u)=e^{u} \cot u $$

Use the Product Rule to find the derivative of each function. \(f(x)=e^{2 x}+2 e^{x}+4\)

Find the derivative of the function. \(f(x)=x^{2.1}\)

Use the definition of the derivative to find the derivative of the function. What is its domain? \(f(x)=3 x^{2}-x+1\)

Find the differential of the function at the indicated number. $$ f(x)=x^{4}-2 x^{3}+3 ; \quad x=0 $$

Identify the "inside function" \(u=f(x)\) and the "outside function" \(y=g(u) .\) Then find \(d y / d x\) using the Chain Rule $$ y=\sec \sqrt{x} $$

Find dy/dx by implicit differentiation. $$ x^{3} y^{2}-2 x^{2} y+2 x=3 $$

Differentiate the function. $$ g(t)=t \ln 2 t $$

In Exercises, \(s(t)\) is the position function of a body moving along a coordinate line; \(s(t)\) is measured in feet and \(t\) in seconds, where \(t \geq 0 .\) Find the position, velocity, and speed of the body at the indicated time. $$ s(t)=t e^{-t} ; \quad t=2 $$

Find the derivative of the function. $$ g(v)=e^{v} \sin v-2 v \csc v $$