3. Techniques of Differentiation

Derivatives Find and simplify the derivative of the following functions.

h(x) = (x−1)(2x²-1) / (x³-1)

Find all points (x, y) on the graph of y = x/(x − 2) with tangent lines perpendicular to the line y = 2x + 3.

Find the derivative of each function.

$y=(3x+5)^2$

Find the derivative of each function.

$f\left(t\right)=2t(t^{-3}+t^{2/3})$

Find the derivative of the function.

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

Find the derivative of the function.

$f\left(x\right)=\frac{2x-1}{x^3+2}$

Find the derivative of the function.

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

{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).

Let a = 3 and find an equation of the line tangent to y = 27 / (x² + 9) at x = 2.

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

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

Evaluate dy/dx and dy/dx|x=2 if y= x+1/x+2

21–30. Derivatives

b. Evaluate f'(a) for the given values of a.

f(x) = 1/x+1; a = -1/2;5

Tangent lines Suppose f(2)=2 and f′(2) =3. Let g(x) = x²f(x) and h(x) = f(x) / x−3.

b. Find an equation of the line tangent to y = h(x) at x=2.

Given that f(1)=2 and f′(1)=2 , find the slope of the curve y=xf(x) at the point (1, 2).

Find the derivative the following ways:

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

f(x) = x(x-1)

Find the derivative the following ways:

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

g(t) = (t + 1)(t² - t + 1)