Find the derivative of the function. h (t) = t3 - 3t
> If f (x) + x2 [f (x)]3 = 10 and f (1) = 2, find f'(1).
> Find dy/dx by implicit differentiation. sin x + cos y = sin x cos y
> Find dy/dx by implicit differentiation. ey cos x = 1 + sin (xy)
> Find dy/dx by implicit differentiation. tan (x – y) = y/1 + xy2
> Find dy/dx by implicit differentiation. ex/y = x - y
> Find dy/dx by implicit differentiation. y sin (x2) = x sin (y2)
> Find dy/dx by implicit differentiation. 4 cos x sin y = 4
> Find dy/dx by implicit differentiation. 1 + x = sin (xy2)
> (a). Find y' by implicit differentiation. (b). Solve the equation explicitly for and differentiate to get y' in terms of x. (c). Check that your solutions to parts (a) and (b) are consistent by substituting the expression for into your solution for part
> Find the derivative of the function. F (x) = √1 - 2x
> Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems. (a). Graph the curve with equation At, how many points does this curve have horizontal tangents? Estimate the -coordinates of these points. (b). Fin
> Find the derivative of the function. F (x) = (4x - x2)100
> Find the derivative of the function. F (x) = (x4 + 3x2 – 2)5
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = √2 - ex
> Find R'(0), R(x) = x + 3x3 + 5x5/1 + 3x3 + 6x6 + 9x9 where Hint: Instead of finding R’(x) first, let f (x) be the numerator and g (x) the denominator of R (x) and compute R' (0) from f (0), f' (0), g (0), and g' (0).
> Let f and be the functions in Exercise 53. Exercise 53: A table of values for f, g, f', and g' is given. (a). If F (x) = f (f (x)), find F'(2). (b). If G (x) = g (g (x)), find G'(3).
> A table of values for f, g, f', and g' is given. (a). If h (x) = f (g (x)), find h'(1). (b). If H (x) = g (f (x)), find H'(1).
> Find the x-coordinates of all points on the curve y = sin 2x – 2 sin x at which the tangent line is horizontal.
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = e√x
> (a). If f (x) = x√2 – x2, find f'(x). (b). Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f'.
> (a). The curve y = |x|/√2 – x2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Suppose that we don’t have a formula for g (x) but we know that g (2) = -4 and g'(x) = √x2 + 5 for all x. (a). Use a linear approximation to estimate g (1.95) and g (2.05). (b). Are your estimates in part (a) too large or too small? Explain.
> (a). Find an equation of the tangent line to the curve y = 2/ (1 + e-x) at the point (0, 1). (b). Illustrate part (a) by graphing the curve and the tangent line on the same screen.
> Find an equation of the tangent line to the curve at the given point. y = sin x + sin2x, (0, 0)
> Find an equation of the tangent line to the curve at the given point. y = sin (sin x), (π, 0)
> Find an equation of the tangent line to the curve at the given point. y = √1 + x3, (2, 3)
> Find y' and y". y = eex
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = sin (cot x)
> Find the derivative of the function. y = √(x+√(x+√x))
> Find the derivative of the function. y = cot2(sin θ)
> Find the derivative of the function. y = sin (sin (sin x))
> Find the derivative of the function. y = 2 simπx
> Suppose that the only information we have about a function f is that f (1) = 5 and the graph of its derivative is as shown. (a). Use a linear approximation to estimate f (0.9) and f (1.1). (b). Are your estimates in part (a) too large or too small? Exp
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx. y = tan πx
> Find the derivative of the function. y = sin (tan 2x)
> Find the derivative of the function. y = ektan√x
> If H (θ) = θ sin θ, find H'(θ) and H"(θ).
> Find the derivative of the function. y = eu – e-u / eu + e-u
> Find the derivative of the function. y = sec2x + tan2x
> Find the derivative of the function. G (y) = (y2/y + 1)5
> Find the derivative of the function. y = (x2 + 1/x2 - 1)3
> Find the derivative of the function. y = 101-x2
> Find the derivative of the function. y = ex cos x
> On page 431 of Physics: Calculus, 2d ed., by Eugene Hecht (Pacific Grove, CA, 2000), in the course of deriving the formula T = 2π√L/g for the period of a pendulum of length L, the author obtains the equation aT = -g sin θ for the tangential acceleration
> Find the derivative of the function. h (t) = (t4 – 1)3 (t3 + 1)4
> Find the derivative of the function. y = (2x – 5)4 (8x2 – 5)-3
> Find the derivative of the function. y = e-2t cos 4t
> Find the derivative of the function. y = xe-kx
> Find the derivative of the function. y = 3 cot (nθ)
> Find the derivative of the function. y = a3 + cos3x
> Find the derivative of the function. y = cos (a3 + x3)
> Find the derivative of the function. f (t) = 3√1 + tan t
> Find the derivative of the function. f (z) = 1/z2 + 1
> Find equations of the tangent line and normal line to the given curve at the specified point. y = 2xex, (0, 0)
> Find the derivative of the function. f (x) = (1 + x4)2/3
> Write the composite function in the form f (g (x)). [Identify the inner function u = g (x) and the outer function y = f (u).] Then find the derivative dy/dx.
> Make a careful sketch of the graph of f and below it sketch the graph of f' in the same manner as in Exercises 4–11. Can you guess a formula for f' (x) from its graph? f (x) = sin x
> The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function M' (t). During which years was the derivative negative?
> Differentiate the function. B (x) = cy-6
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 6. if limx→ 6 [f (x) g (x)] exists, then the limit must be f (6) g (6). 7. I
> Determine whether the statement is true or false. If it is true, explain why If it is false, explain why or give an example that disproves the statement. 1. If f and g are differentiable, then 2. If f and g are differentiable, then 3. If f and g are
> Differentiate. y = x/2 – tan x
> Differentiate the function. h (x) = (x – 2) (2x + 3)
> Use the given graph to estimate the value of each derivative. Then sketch the graph of f'. (a). f'(-3) (b). f'(-2) (c). f'(-1) (d). f'(0) (e). f'(1) (f). f'(2) (g). f'(3)
> If R denotes the reaction of the body to some stimulus of strength x, the sensitivity is defined to be the rate of change of the reaction with respect to x. A particular example is that when the brightness x of a light source is increased, the eye reacts
> On what interval is the function f (x) = 5x - ex increasing?
> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = x + 1/x
> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = 3x15 - 5x3 + 3
> Find f'(x). Compare the graphs of f and f' and use them to explain why your answer is reasonable. f (x) = ex - 5x
> Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = x - √x (1, 0)
> Differentiate. f (t) = cot t/et
> Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = 3x2 - x3 (1, 2)
> Find equations of the tangent line and normal line to the curve at the given point. y = x4 + 2e (0, 2)
> Find an equation of the tangent line to the curve at the given point. y = x4 + 2x2 – x, (1, 2)
> Find an equation of the tangent line to the curve at the given point. y = 4√x, (1, 1)
> For what values of does the graph of have a horizontal tangent? f (x) = x + 2 sin x
> Differentiate the function. y = ex+1 + 1
> Differentiate the function. z = A/y10 + Bey
> Differentiate the function. v = (√x + 1/√x)2
> Evaluate limx→1 x1000 – 1/x - 1
> Differentiate the function. u = 5√t + 4√t5
> Differentiate. y = c cos t+ t2 sin t
> Differentiate the function. y = aev + b/v + c/v2
> Find the value of such that the line y = 3/2 + 6x is tangent to the curve y = c√x.
> Differentiate the function. y = 4π2
> Differentiate the function. g (u) = √2 u + √3 u
> (a). The curve with equation y2 = x3 – 3x2 is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point (1, -2). (b). At what points does this curve have horizontal tangents? (c). Illustrate parts (a) and (b) by grap
> The graph indicates how Australia’s population is aging by showing the past and projected percentage of the population aged 65 and over. Use a linear approximation to predict the percentage of the population that will be 65 and over in
> Differentiate the function. y = x2 + 4x +3/√x
> Use the result of Exercise 63(c) to find an antiderivative of each function. (a). f (x) = √x (b). f (x) = ex + 8x3 Exercise 63(c): (c). Find an antiderivative for f (x) xn, where n ≠ -1. Check by differentiation.
> Differentiate the function. f (x) = x2 -3x + 1/x2
> The equation y" + y' -2y = x2 is called a differential equation because it involves an unknown function and its derivatives y' and y". Find constants A, B, and C such that the function y = Ax2 + BX + C satisfies this equation. (Differential equations wil
> Find a second-degree polynomial P such that P (2) = 5, P'(2) = 3, and P"(2) = 2.
> (a). If g is differentiable, the Reciprocal Rule says that Use the Quotient Rule to prove the Reciprocal Rule. (b). Use the Reciprocal Rule to differentiate the function in Exercise 16. Exercise 16: Differentiate y = 1/s + kes (c). Use the Reciproc
> Differentiate. g (θ) = eθ (tan θ – θ)
> Use the definition of a derivative to show that if f (x) = 1/x, then f'(x) = -1/x2. (This proves the Power Rule for the case n = -1.)
> (a). If F (x) = f (x) g (x), where f and g have derivatives of all orders, show that F" = f"g +2f'g' + fg". (b). Find similar formulas for f"' and F(4). (c). Guess a formula for F(R).
> (a). Use the Product Rule twice to prove that if f, g, and h are differentiable, then (fgh)' = f'gh + fg'h + fgh'. (b). Taking f = g = h in part (a), show that d/ dx [f (x)]3 = 3[f (x)]2 f'(x) (c). Use part (b) to differentiate y = e3x.
> The cost, in dollars, of producing yards of a certain fabric is C (x) = 1200 + 12x – 0.1x2 + 0.0005x3 (a). Find the marginal cost function. (b). Find C"(200) and explain its meaning. What does it predict? (c). Compare C"(200) with the cost of manufacturi
> Use the method of Exercise 55 to compute Q'(0), where Q (x) = 1 + x + x2 + xex/ 1 – x + x2 - xex Exercise 55: Find R'(0), R(x) = x + 3x3 + 5x5/1 + 3x3 + 6x6 + 9x9 where Hint: Instead of finding R’(x) first, let f (x) be the numerator and g (x) the deno
