3. Techniques of Differentiation

Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>

d. d/dx (f(x)³) |x=5

27–76. Calculate the derivative of the following functions.

$y={(3x^2+7x)}^{10}$

Use the given graphs of f and g to find each derivative. <IMAGE>

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

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

𝔂 = 3 .

(5x² + sin 2x)³/²

The Chain Rule for second derivatives

b. Use the formula in part (a) to calculate $\frac{d^2}{d{x}^2}\left(\sin (3x^4+5x^2+2)\right)$.

Find the derivatives of the functions in Exercises 19–40.

y = (1 / 18)(3x − 2)⁶ + (4 − (1 / 2x²))⁻¹

{Use of Tech} Tangent lines Determine equations of the lines tangent to the graph of y= x√5−x² at the points (1, 2) and (−2,−2). Graph the function and the tangent lines.

Find the derivative of the function.

$f(x)=\(\sqrt{(5x^2-3x)}\)$

Derivatives by different methods

a. Calculate d/dx (x²+x)² using the Chain Rule. Simplify your answer.

27–76. Calculate the derivative of the following functions.

$y=\sqrt{10x+1}$

Falling meteorite The velocity of a heavy meteorite entering Earth’s atmosphere is inversely proportional to √s when it is s km from Earth’s center. Show that the meteorite’s acceleration is inversely proportional to s².

Deriving trigonometric identities

b. Verify that you obtain the same identity for sin2t as in part (a) if you differentiate the identity cos 2t = 2 cos² t−1.

5–24. For each of the following composite functions, find an inner function u=g(x) and an outer function y=f(u) such that y=f(g(x)). Then calculate dy/dx.

y = tan 5x²

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

$\underset{x\to 2}{\lim }\frac{{(x^2-3)}^5-1}{x-2}$

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.

59. y = √(t/(t+1))