3. Techniques of Differentiation

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.

Additional Applications

Bacterium population

When a bactericide was added to a nutrient broth in which bacteria were growing, the bacterium population continued to grow for a while, but then stopped growing and began to decline. The size of the population at time t (hours) was b = 10⁶ + 10⁴t − 10³t². Find the growth rates at

a. t = 0 hours.

b. t = 5 hours.

c. t = 10 hours.

Vehicular stopping distance Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is s = 1.1v + 0.054v², where s is measured in ft and v in mph. The linear term 1.1v models the distance the car travels during the time the driver perceives a need to stop until the brakes are applied, and the quadratic term 0.054v² models the additional braking distance once they are applied. Find ds/dv at v = 35 and v = 70 mph, and interpret the meaning of the derivative.