Find and simplify the derivative of the following functions.

f(x) = e^x(x³ − 3x² + 6x − 6)

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.)

Product Rule for three functi​​ons Assume f, g, and h are differentiable at x.

b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

f(w) = w³ -w / w

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

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

y = (x + 5) / (x - 1); a = 3

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

y = (x + 5) / (x - 1); a = 3

Derivatives Find and simplify the derivative of the following functions.

g(x) = e^x / x²-1

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

Derivatives Find and simplify the derivative of the following functions.

g(w) = √w+w / √w-w

Derivatives Find and simplify the derivative of the following functions.

h(w) = w⁵/³ / w⁵/³+1

Derivatives Find and simplify the derivative of the following functions.

g(x) = x⁴/³-1 / x⁴/³+1

Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).

f(x) = 1/x

Given that f(1) = 5, f′(1) = 4, g(1) = 2, and g′(1) = 3 , find d/dx (f(x)g(x))∣ ∣x=1 and d/dx (f(x) / g(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).

Derivatives Find and simplify the derivative of the following functions.

f(x) = x /x+1

Derivatives Find and simplify the derivative of the following functions.

f(x) = 2e^x-1 / 2e^x+1

Derivatives Find and simplify the derivative of the following functions.

s(t) = t⁴/³ / e^t

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = √3x; a= 12

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = √3x; a= 12

Derivatives and tangent lines

b. Determine an equation of the line tangent to the graph of f at the point (a,f(a)) for the given value of a.

f(x) = 1/3x-1; a= 2

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

a. Compute A'(t). What units are associated with this derivative and what does the derivative measure?

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

b. How fast will the city be growing when it reaches a size of 38 mi²?

City urbanization City planners model the size of their city using the function A(t) = - 1/50t² + 2t +20, for 0 ≤ t ≤ 50, where A is measured in square miles and t is the number of years after 2010.

c. Suppose the population density of the city remains constant from year to year at 1000 people mi². Determine the growth rate of the population in 2030.

Derivative calculations Evaluate the derivative of the following functions at the given point.

f(s) = 2√s-1; a=25

Find an equation of the line tangent to the following curves at the given value of x.

y = 4 sin x cos x; x = π/3

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers. 

y = 12s³-8s²+12s/4s

Find an equation of the line tangent to the curve y = sin x at x = 0.

Find d²/dx² (sin x + cos x).