Find y'' for the following functions.

y = e^x sin x

Evaluate the derivative of the following functions.

f(x) = sin^-1 2x

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?

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

Find d/dx(ln√x²+1).

​​​​​47​–​56.​ Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x)=tan x; (1,π/4)

Evaluate the derivative of the following functions.

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

Find the derivative of the following functions.

y = x² (1 - In x²)

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

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

h(x) = √x (√x-x³/²)

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

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

y = x⁵

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.

A projectile is fired vertically upward into the air; its position (in feet) above the ground after t seconds is given by the function s (t). For the following functions, use limits to determine the instantaneous velocity of the projectile at t = a seconds for the given value of a.

s(t) = -16t² + 128t + 192; a = 2

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

f(x) = x^In x

Find an equation of the line tangent to the curve y = sin x at x = 0.

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

13​–​40.​ Evaluate the derivative of the following functions.

f(x) = 1/tan^−1(x²+4)

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

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).

y = 4 log₃(x²−1)

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

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

y = 10^In 2x

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.

9–61. Evaluate and simplify y'.

y = (2x−3)x^3/2

A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.

y = (x²+1)^x

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>

5–8. Calculate dy/dx using implicit differentiation.

x = y²

Continuity of a piecewise funct​​ion Let g(x) = <matrix 2x1> For what values of a is g continuous?