3. Techniques of Differentiation

Evaluate and simplify y'.

y = 2x^√2

{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).

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.)

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.

g(t) = t³+3t²+t / t³

State the derivative rule for the logarithmic function f(x)=log(subscript b)x. How does it differ from the derivative formula for ln x?

Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.

Derivatives Find and simplify the derivative of the following functions.

f(x) = 3x⁴(2x²−1)

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

𝔂 = x⁵ - 0.125x² + 0.25x

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

𝔂 = x³ - 3 (x² + π²)

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

a. 6ƒ(x) - g(x), x = 1

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.

Slopes and Tangent Lines

b. Smallest slope What is the smallest slope on the curve? At what point on the curve does the curve have this slope?

Quadratics having a common tangent line The curves y = x² + ax + b and y = cx − x² have a common tangent line at the point (1,0). Find a, b, and c.

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.