Explain the difference between the average rate of change and the instantaneous rate of change of a function f.

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→a (mx+b)=ma+b, for any constants a, b, and m

A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?

Find the derivative of the following functions.

y = e^-x sin x

Con​​sider the curve x=e^y. Use implicit differentiation to verify that dy/dx = e^-y and then find d²y/dx² .

Verifying derivative formulas Verify the following derivative formulas using the Quotient Rule.

d/dx (csc x) = -csc x cot x

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

Find y'' for the following functions.

y = cos θ sin θ

Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.

49–55. Derivatives of tower functions (or g^h) Find the derivative of each function and evaluate the derivative at the given value of a.

h (x) = x^√x; a = 4

Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).

f(x) = In(sec⁴x tan² x)

If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at ​ x=1, 2, and 3​​​.

27–76. Calculate the derivative of the following functions.

$\sqrt[3]{x^2+9}$

Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.

(lim h🠂0) (2+h)⁴-16 / h

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = x^In x

15–48. Derivatives Find the derivative of the following functions.

y = 10^x(In 10^x-1)

15–48. Derivatives Find the derivative of the following functions.

P = 40/1+2^-t

Find the derivative of the following functions.

y = In √x⁴+x²

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.

√x⁴+y² = 5x+2y³

15–48. Derivatives Find the derivative of the following functions.

y = 10^In 2x

Angle to a particle (part 2) The figure in Exercise 81 shows the particle traveling away from the sensor, which may have influenced your solution (we expect you used the inverse sine function). Suppose instead that the particle approaches the sensor (see figure). How would this change the solution? Explain the differences in the two answers. <IMAGE>

Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)

lim x→π/2 cos x/x−π/2

Find d/dx(ln√x²+1).

A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.

At all times, the length of the long leg of a right triangle is 3 times the length x of the short leg of the triangle. If the area of the triangle changes with respect to time t, find equations relating the area A to x and dA/dt to dx/dt.

75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).

f(x) = tan¹⁰x / (5x+3)⁶

{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?

f(x) = sin x on [−π,π]

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

Evaluate the derivative of the following functions.

f(x) = sin^-1 (e^-2x)