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

f(v) = v¹⁰⁰+e^v+10

Find the derivative of the following functions.

y = x² (1 - In x²)

Suppose w(t) is the weight (in pounds) of a golden retriever puppy t weeks after it is born. Interpret the meaning of w'(15) = 1.75.

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

67​–​78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.

f(x) = e^3x+1

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

$y={(x^2+2x+7)}^8$

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

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

y = 10^x(In 10^x-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)=4e^10x; (4,0)

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 5x) / x

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

y = 4^-x sin x

If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.

Derivative calculations Evaluate the derivative of the following functions at the given point.

f(s) = 2√s-1; a=25

Find d/dx(ln√x²+1).

51–56. Second derivatives Find d²y/dx².

x⁴+y⁴ = 64

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?

A surface ship is moving (horizontally) in a straight line at 10 km/hr. At the same time, an enemy submarine maintains a position directly below the ship while diving at an angle that is 20° below the horizontal. How fast is the submarine’s altitude decreasing?

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (tan 7x) / (sin x)

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

r(t) = (e^2t + 3e^t + 2) / (e^t + 2)

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

P = 40/1+2^-t

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

y = In (x³+1)^π

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 = (cos x) In cos²x

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

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

Tangent lines Find an equation of the line tangent to the graph of f at the given point.

f(x) = sec⁻¹(e^x); (ln 2,π/3)

Simplify the expression e^xln(x²+1).

Find the derivative of the following functions.

y = x² In x

Find the derivative of the following functions.

y = In x / (In x + 1)

The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?

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(3x + 1)⁴