3. Techniques of Differentiation

Population growth Consider the following population functions.

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

p(t) = 600 (t²+3/t²+9)

Find the derivative of the function.

$g\left(y\right)=\frac{5y^2+2y-1}{y^2-2}$

Find and simplify the derivative of the following functions.

h(x) = (5x⁷ + 5x)(6x³ + 3x² + 3)

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

y = 2x² / (3x - 1); a = 1

Derivatives from a table Use the following table to find the given derivatives. <IMAGE>

d/dx (f(x)g(x)) |x=1

15–48. Derivatives Find the derivative of the following functions.

y = 4^-x sin 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.

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

f(t) = 1/t+1; a=1

Suppose the line tangent to the graph of f at x=2 is y=4x+1 and suppose y=3x−2 is the line tangent to the graph of g at x=2. Find an equation of the line tangent to the following curves at x=2.

y = f(x)g(x)

Find the derivative of the function.

$y=\frac{2-3t}{4t^2+7}$

Find the derivative the following ways:

Using the Product Rule or the Quotient Rule. Simplify your result.

f(x) = (x - 1)(3x + 4)

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

𝔂 = x⁻¹/² sec (2x)²

Derivatives of sin^n x Calculate the following derivatives using the Product Rule.

c. d/dx (sin⁴ x)

Product Rule for three functions Assume f, g, and h are differentiable at x.

a. Use the Product Rule (twice) to find a formula for d/dx (f(x)g(x)h(x)).

Derivatives Find and simplify the derivative of the following functions.

s(t) = t⁴/³ / e^t