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

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

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

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

73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>

Exercise 46

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

Find the derivative of the following functions.

y = In |sin x|

Evaluate the derivative of the following functions.

f(x) = sin^-1 2x

Evaluate the derivative of the following functions.

f(x) = sin(tan^-1 (ln x))

Find the derivative of the following functions.

y = x² In x

Explain why b^x = e^xlnb.

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

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

f(x) = x⁸cos³ x / √x-1

Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).

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.

23–51. Calculating derivatives Find the derivative of the following functions.

y = a sin x + b cos x/a sin x - b cos x; a and b are nonzero constants

Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.

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

f(s) = √s/4

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.)

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

6x³+7y³ = 13xy

Evaluate the derivative of the following functions.

f(y) = tan^-1 (2y² - 4)

{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.

Calculator limits Use a calculator to approximate the following limits.

lim x🠂0 e^3x-1 / x

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

Given that f'(3) = 6 and g'(3) = -2 find (f+g)'(3).

Derivatives Find and simplify the derivative of the following functions.

s(t) = t⁴/³ / e^t

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

Find y'' for the following functions.

y = cos θ sin θ