Find and simplify the derivative of the following functions.

h(x) = (5x⁷ + 5x)(6x³ + 3x² + 3)

Calculate the derivative of the following functions.

y = (1 - e^0.05x)^-1

Determining the unknown constant Let f(x) = {2x² if x≤1 ax-2 if x>1. Determine a value of a (if possible) for which f' is continuous at x=1.

{Use of Tech} Equations of tangent lines

Find an equation of the line tangent to the given curve at a.

y = −3x² + 2; a=1

{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 = −3x²+2; a=1

Calculate the derivative of the following functions.

y = √x+√x+√x

{Use of Tech} Equations of tangent lines

Find an equation of the line tangent to the given curve at a.

y = e^x; a = ln 3

{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

Calculate the derivative of the following functions.

y = (f(g(x^m)))^n, where f and g are differentiable for all real numbers and m and n are constants

Find an equation of the line tangent to the given curve at a.

y = 2x² / (3x - 1); a = 1

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

y = 2x² / (3x - 1); a = 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=3x−4; P(1, −1)

Let f(x) = x² - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 0.

Let f(x) = x² - 6x + 5.

Find the values of x for which the slope of the curve y = f(x) is 2.

Calculate the derivative of the following functions.

y = (e^x / x+1)⁸

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 = 2/x; P(1, 2)

Calculate the derivative of the following functions.

y = tan(xe^x)

Population growth Consider the following population functions.

a. Find the instantaneous growth rate of the population, for t≥0.

p(t) = 600 (t²+3/t²+9)

Population growth Consider the following population functions.

b. What is the instantaneous growth rate at t=5?

p(t) = 600 (t²+3/t²+9)

Population growth Consider the following population functions.

c. Estimate the time when the instantaneous growth rate is greatest.

p(t) = 600 (t²+3/t²+9)

{Use of Tech} Population growth Consider the following population functions.

d. Evaluate and interpret lim t→∞ p(t).

p(t) = 600 (t²+3/t²+9)

Population growth Consider the following population functions.

e.Use a graphing utility to graph the population and its growth rate.

p(t) = 600 (t²+3/t²+9)

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x)=x²+1; Q(3, 6)

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line is horizontal.

Let f(x) = 4√x - x.

Find all points on the graph of f at which the tangent line has slope -1/2.

Find f′(x), f′′(x), and f′′′(x) for the following functions.

f(x) = 3x³ + 5x² + 6x

Given the function f and the point Q, find all points P on the graph of f such that the line tangent to f at P passes through Q. Check your work by graphing f and the tangent lines.

f(x) = 1/x; Q (-2, 4)

The following equations implicitly define one or more functions.

a. Find dy/dx using implicit differentiation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

The following equations implicitly define one or more functions.

b. Solve the given equation for y to identify the implicitly defined functions y=f₁(x), y = f₂(x), ….

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)

The following equations implicitly define one or more functions.

c. Use the functions found in part (b) to graph the given equation.

x+y³−xy=1 (Hint: Rewrite as y³−1=xy−x and then factor both sides.)