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

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

Find y'' for the following functions.

y = tan x

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

s(t) = 4√t - 1/4t⁴+t+1

Find f′(1) when f(x) = x^(1/x).

Evaluate and simplify y'.

y = 2x^√2

Evaluate and simplify y'.

y = 4u²+u / 8u+1

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

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

66–71. Higher-order derivatives Find and simplify y''.

x + sin y = y

Evaluate and simplify y'.

y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰

9–61. Evaluate and simplify y'.

y = e^tan x (tan x−1)

9–61. Evaluate and simplify y'.

y = tan (sin θ)

Evaluate and simplify y'.

y = ln w / w⁵

9–61. Evaluate and simplify y'.

y = e^2θ

9–61. Evaluate and simplify y'.

y = e^6x sin x

Evaluate and simplify y'.

y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷

9–61. Evaluate and simplify y'.

y = x^√x+1

A jet flying at 450 mi/hr and traveling in a straight line at a constant elevation of 500 ft passes directly over a spectator at an air show. How quickly is the angle of elevation (between the ground and the line from the spectator to the jet) changing 2 seconds later? 

9–61. Evaluate and simplify y'.

y = e^sin x+2x+1

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

Evaluate and simplify y'.

y = ln |sec 3x|

9–61. Evaluate and simplify y'.

y = e^sin (cosx)

9–61. Evaluate and simplify y'.

y = x²+2x tan^−1(cot x)

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

c. d/dx ((f(x) / g(x)) |x=3

Evaluate d/dx(x sec^−1 x) |x = 2 /√3.

Higher-order derivatives Find and simplify y''.

y = 2^x x

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

b. d/dx ((f(x) / g(x)) |x=

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.

9–61. Evaluate and simplify y'.

y = tan^−1 √t²−1

Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

y =√x³+x−1 at y=3

9–61. Evaluate and simplify y'.

y = 10^sin x+sin¹⁰x