3. Techniques of Differentiation

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

85. y = log₂(8t^(ln 2))

a. Find an equation for the line that is tangent to the curve y = x³ − 6x² + 5x at the origin.

[Technology Exercise] b. Graph the curve and tangent line together. The tangent line intersects the curve at another point. Use Zoom and Trace to estimate the point’s coordinates.

[Technology Exercise] c. Confirm your estimates of the coordinates of the second intersection point by solving the equations for the curve and tangent line simultaneously.

{Use of Tech} Equations of tangent lines

b. Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = e^x; a = ln 3

Derivatives from a graph Let F = f + g and G = 3f - g, where the graphs of f and g are shown in the figure. Find the following derivatives. <IMAGE>

F'(2)

Find the indicated derivative.

$f(x)=3x-4$

$f^{\(\prime\)}(x)=$

Evaluate and simplify y'.

y = 2x^√2

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

73. y = log₄ x + log₄ x²

Slopes and Tangent Lines

a. Horizontal tangent lines Find equations for the horizontal tangent lines to the curve y = x³ − 3x − 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.

Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at

c. x=3

Find the indicated derivative.

$h\left(t\right)=\frac{1}{2}{t}^3+\frac{4}{t^2}+3\sqrt{t}$

$h^{\prime }\left(t\right)=$

Find the derivatives of the functions in Exercises 1–42.

𝔂 = x⁵ - 0.125x² + 0.25x

{Use of Tech} The Witch of Agnesi The graph of y = a³ / x²+a², where a is a constant, is called the witch of Agnesi (named after the 18th-century Italian mathematician Maria Agnesi).

b. Plot the function and the tangent line found in part (a).

In Exercises 59–86, find the derivative of y with respect to the given independent variable.

61. y = 5√s

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)

Which of the following statements about differentiation is true?