1 Functions And Models 2 Limits And Derivatives 3 Differentiation Rules 4 Applications Of Differentiation 5 Integrals 6 Applications Of Integration 7 Techniques Of Integration 8 Further Applications Of Integration 9 Differential Equations 10 Parametric Equations And Polar Coordinates 11 Infinite Sequences And Series 12 Vectors And The Geometry Of Space 13 Vector Functions 14 Partial Derivatives 15 Multiple Integrals 16 Vector Calculus 17 Second-order Differential Equations expand_more
3.1 Derivatives Of Polynomials And Exponential Functions 3.2 The Product And Quotient Rules 3.3 Derivatives Of Trigonometric Functions 3.4 The Chain Rule 3.5 Implicit Differentiation 3.6 Derivatives Of Logarithmic Functions 3.7 Rates Of Change In The Natural And Social Sciences 3.8 Exponential Growth And Decay 3.9 Related Rates 3.10 Linear Approximations And Differentials 3.11 Hyperbolic Functions Chapter Questions expand_more
Problem 1E: Find the derivative of f(x) = (1 + 2x2)(x x2) in two ways: by using the Product Rule and by... Problem 2E: Find the derivative o f the function F(x)=x45x3+xx2 in two ways: by using the Quotient Rule and by... Problem 3E: Differentiate. f(x) = (3x2 5x)ex Problem 4E: 4. Differentiate. g(x)=(x+2x)ex Problem 5E: Differentiate. y=xex Problem 6E: Differentiate. y=ex1ex Problem 7E: Differentiate. g(x)=1+2x34x Problem 8E: Differentiate. G(x)=x222x+1 Problem 9E: Differentiate. H(u)=(uu)(u+u) Problem 10E: Differentiate. J(v) = (v3 2v)(v4 + v2) Problem 11E Problem 12E: Differentiate. f(z) = (1 ez)(z + ez) Problem 13E: Differentiate. y=x2+1x31 Problem 14E Problem 15E: Differentiate. y=t3+3tt24t+3 Problem 16E: Differentiate. y=1t3+2t21 Problem 17E: Differentiate. y=ep(p+pp) Problem 18E: Differentiate. h(r)=aerb+er Problem 19E: Differentiate. y=sss2 Problem 20E Problem 21E: Differentiate. f(t)=t3t3 Problem 22E: Differentiate. V(t)=4+ttet Problem 23E: Differentiate. f(x)=x2exx2+ex Problem 24E Problem 25E: Differentiate. f(x)=xx+cx Problem 26E: Differentiate. f(x)=ax+bcx+d Problem 27E: Find f'(x) and f"(x). f(x) = (x3 + 1)ex Problem 28E: Find f'(x) and f"(x). f(x)=xex Problem 29E: Find f'(x) and f"(x). f(x)=x21+ex Problem 30E: Find f'(x) and f"(x). f(x)=xx21 Problem 31E: Find an equation of the tangent line to the given curve at the specified point. y=x21x2+x+1,(1,0) Problem 32E: Find an equation of the tangent line to the given curve at the specified point. y=1+x1+ex,(0,12) Problem 33E: Find equations of the tangent line and normal line to the given curve at the specified point. y =... Problem 34E: Find equations of the tangent line and normal line to the given curve at the specified point.... Problem 35E: (a) The curve y = 1/(1 + x2) is called a witch of Maria Agnesi. Find an equation of the tangent line... Problem 36E: (a) The curve y = x/(1 + x2) is called a serpentine. Find an equation of the tangent line to this... Problem 37E: (a) If f(x) = (x3 x)ex, find f'(x). (b) Check to see that your answer to part (a) is reasonable by... Problem 38E Problem 39E: (a) If f(x) = (x2 1)/(x2 + 1), find f'(x) and f"(x). (b) Check to see that your answers to part (a)... Problem 40E: (a) If f(x) = (x2 1)ex, find f'(x) and f"(x). (b) Check to see that your answers to part(a) are... Problem 41E: If f(x) = x2/(l + x), find f"(1). Problem 42E: If g(x) = x/ex. find g(n)(x). Problem 43E: Suppose that f(5) = 1, f'(5) = 6, g(5) = 3, and g'(5) = 2. Find the following values. (a) (fg)'(5)... Problem 44E: Suppose that f(4) = 2, g(4) = 5, f'(4) = 6. and g'(4) = 3. Find h'(4). (a) h(x) = 3f(x) + 8g(x) (b)... Problem 45E: If f(x) = exg(x), where g(0) = 2 and g'(0) = 5, find f'(0). Problem 46E: If h(2) = 4 and h'(2) = 3, find ddx(h(x)x)|x=2 Problem 47E: If g(x) = xf(x), where f(3) = 4 and f'(3) = 2, find an equation of the tangent line to the graph of... Problem 48E: If f(2) = 10 and f'(x) = x2f(x) for all x, find f"(2). Problem 49E: If f and g are the functions whose graphs are shown, let u(x) = f(x)g(x) and v(x) = f(x)/g(x). (a)... Problem 50E: Let P(x) = F(x)G(x) and Q(x) = F(x)/G(x), where F and G are the functions whose graphs are shown.... Problem 51E: If g is a differentiable function, find an expression for the derivative of each of the following... Problem 52E: If f is a differentiable function, find an expression for the derivative of each of the following... Problem 53E: How many tangent lines to the curve y = x/(x + 1) pass through the point (1, 2)? At which points do... Problem 54E: Find equations of the tangent lines to the curve y=x1x+1 that are parallel to the line x 2y = 2. Problem 55E: Find R'(0), where R(x)=x3x3+5x51+3x3+6x6+9x9 Hint: Instead of finding R'(x) first, let f(x) be the... Problem 56E: Use the method of Exercise 55 to compute Q'(0), where Q(x)=1+x+x2+xex1x+x2xex Problem 57E: In this exercise we estimate the rate at which the total personal income is rising in the... Problem 58E: A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric... Problem 59E: The Michaelis-Menten equation for the enzyme chymotrypsin is v=0.14[S]0.015+[S] where v is the rate... Problem 60E Problem 61E: (a) Use the Product Rule twice to prove that if f, g, and h are differentiable, then (fgh)' = f'gh +... Problem 62E: (a) If F(x) = f(x) g(x), where f and g have derivatives of all orders, show that F" = f"g + 2f'g' +... Problem 63E Problem 64E: (a) If g is differentiable, the Reciprocal Rule says that ddx[1g(x)]=g(x)[g(x)]2 Use the Quotient... format_list_bulleted