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.

The line tangent to the graph of f at x=5 is y = 1/10x-2. Find d/dx (4f(x)) |x+5

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)

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.

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

f(x) = x^In x

Find the derivative of the following functions.

y = e^-x sin x

90–93. {Use of Tech} Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure that they actually lie on the curve. Confirm your results with a graph.

x²(3y²−2y³) = 4

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

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

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)

Evaluate the derivative of the following functions.

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

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

y = x⁵

Use implicit differentiation to find dy/dx.

e^xy = 2y

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)

The speed of sound (in m/s) in dry air is approximated the function v(T) = 331 + 0.6T, where T is the air temperature (in degrees Celsius). Evaluate v' (T) and interpret its meaning.

Derivatives Find and simplify the derivative of the following functions.

g(t) = 3t² + 6/t⁷

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

Find d²/dx² (sin x + cos x).

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

h(t) = t²/2 + 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

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

y = 4^-x sin x

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

f(x) = tan¹⁰x / (5x+3)⁶

Two boats leave a port at the same time, one traveling west at 20 mi/hr and the other traveling southwest ( 45° south of west) at 15 mi/hr. After 30 minutes, how far apart are the boats and at what rate is the distance between them changing? (Hint: Use the Law of Cosines.)

Find and simplify the derivative of the following functions.

f(x) = e^x(x³ − 3x² + 6x − 6)

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

f(x) = x⁸cos³ x / √x-1

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

An object dropped from rest falls d(t)=16t² feet in t seconds. Find d′(4).

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).

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

​​​​​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)