Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

Derivatives Find and simplify the derivative of the following functions.

f(t) = t⁵/³e^t

​​​​​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) = 1/2x+8; (10,4)

Find y'' for the following functions.

y = e^x sin x

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

f(x) = x⁸cos³ x / √x-1

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.

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

sin y+2 = x

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

Explain why b^x = e^xlnb.

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.

g (x) = x^ In x; a = e

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

e^y-e^x = C, where C is constant

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)

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 x🠂2) 1/x+1 - 1/3 / x-2

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

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

Find and simplify the derivative of the following functions.

f(x) = √(e^2x + 8x²e^x +16x⁴) (Hint: Factor the function under the square root first.)

Use limits to find f' (x) if f(x) = 7x.

9–61. Evaluate and simplify y'.

y = (v / v+1)^4/3

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

A line perpendicular to another line or to a tangent line is often called a normal line. Find an equation of the line perpendicular to the line that is tangent to the following curves at the given point P.

y= √x; P(4, 2)

Find the derivative of the following functions.

y = x² (1 - In x²)

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

y = 4^-x sin x

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.

{Use of Tech} Tangent line Find the equation of the line tangent to y=2^sin x at x=π/2. Graph the function and the tangent line.

Piston compression A piston is seated at the top of a cylindrical chamber with radius 5 cm when it starts moving into the chamber at a constant speed of 3 cm/s (see figure). What is the rate of change of the volume of the cylinder when the piston is 2 cm from the base of the chamber? <IMAGE>

Find the slope of the line tangent to the graph of y = tan^−1 x at x= −2.

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.

s(t) = 4√t - 1/4t⁴+t+1

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

y = (x² - 2ax + a²) / (x - a); a is a constant.

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