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Problem 3.10.57
Find (f^−1)′(3), where f(x)=x³+x+1.
Problem 3.9.13
Find d/dx (In(xe^x)) without using the Chain Rule and the Product Rule.
Problem 3.25
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
h(t) = t²/2 + 1
Problem 3.R.65
Find f′(1) when f(x) = x^(1/x).
Problem 3.R.12
Evaluate and simplify y'.
y = 2x√2
Problem 3.R.35
Evaluate and simplify y'.
y = 4u²+u / 8u+1
Problem 3.R.33
Evaluate and simplify y'.
y = ln w / w⁵
Problem 3.R.70
66–71. Higher-order derivatives Find and simplify y''.
x + sin y = y
Problem 3.R.84b
Use the given graphs of f and g to find each derivative. <IMAGE>
b. d/dx (f(x)g(x)) |x=1
Problem 3.R.61
Evaluate and simplify y'.
y = (3x+5)¹⁰ √x²+5 / (x³+1)⁵⁰
Problem 3.R.37
9–61. Evaluate and simplify y'.
y = tan (sin θ)
Problem 3.R.26
9–61. Evaluate and simplify y'.
y = e^sin x+2x+1
Problem 3.R.60
Evaluate and simplify y'.
y = (x²+1)³ / (x⁴+7)⁸(2x+1)⁷
Problem 3.R.28
9–61. Evaluate and simplify y'.
y = e^tan x (tan x−1)
Problem 3.R.46
9–61. Evaluate and simplify y'.
y = e^6x sin x
Problem 3.R.13
9–61. Evaluate and simplify y'.
y = e^2θ
Problem 3.R.30
9–61. Evaluate and simplify y'.
y = x^√x+1
Problem 3.R.73
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)
Problem 3.R.43
9–61. Evaluate and simplify y'.
y = x²+2x tan^−1(cot x)
Problem 3.R.84C
Use the given graphs of f and g to find each derivative. <IMAGE>
c. d/dx ((f(x) / g(x)) |x=3
Problem 3.R.67
Higher-order derivatives Find and simplify y''.
y = 2^x x
Problem 3.R.96b
{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.
Problem 3.R.40
9–61. Evaluate and simplify y'.
y = e^sin (cosx)
Problem 3.R.29
9–61. Evaluate and simplify y'.
y = tan^−1 √t²−1
Problem 3.R.10
Evaluate and simplify y'.
y = 4x⁴ ln x − x⁴
Problem 3.R.63
Evaluate d/dx(x sec^−1 x) |x = 2 /√3.
Problem 3.R.47
9–61. Evaluate and simplify y'.
y = 2^x²−x
Problem 3.R.48
9–61. Evaluate and simplify y'.
y = 10^sin x+sin¹⁰x
Problem 3.R.87
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
Problem 3.R.17
9–61. Evaluate and simplify y'.
y = 5t² sin t