Chapter 4 Review

Chapter 4: Short-cuts to differentiation



• SECTION 4.1: Derivative formulas for powers and polynomials
  
  
  1. Suggested Problems: #14, 20, 24, 29, 32, 39.
     
  2. The derivative of a constant is 0:
     d/dx(c) = 0, where c is constant.
     
  3. The derivative of a linear function f(x) = m x + b is f(x) = m, the slope of the line that it represents.
     
  4. The derivative of a constant times a function is the constant times the derivative of the function: If f(x) is differentiable, then
     d/dx ( c f(x) ) = c f(x), where c is constant.
     
  5. The derivative of a sum or difference of functions is the sum or difference of derivatives of the functions: If f(x) and g(x) are differentiable functions, then
     d/dx ( f(x) ± g(x) ) = f(x) ± g(x).
  6. The derivative of a power of x:
     d/dx ( x^r ) = r x^r-1, where r is a constant.
     
  7. The derivative of a polynomial:
     d/dx ( a_n xⁿ + a_n-1 x^n-1 + ··· + a₂ x² + a₁ x + a₀ ) = n a_n x^n-1 + (n - 1) a_n-1 x^n-2 + ··· + 2 a₂ x + a₁.
  8. The slope of the line tangent to the graph of the curve y = f(x) at the point (a, f(a)) is given by f(a), so that the equation of the line tangent to the curve at this point is y - f(a) =f(a) (x - a).
     
  
  
  
• SECTION 4.2: Exponential and logarithmic functions
  
  
  1. Suggested Problems: #2, 3, 6, 12, 16, 24, 30.
     
  2. The derivative of the exponential function with base e:
     d/dx ( e^x ) = e^x.
  3. The derivative of the exponential function with base a, where a > 0, a 1:
     d/dx ( a^x ) = ln(a) · a^x.
  4. The derivative of the natural logarithm function:
     d/dx ( ln(x) ) = 1/x.
  
  
  
• SECTION 4.3: The chain rule
  
  
  1. Suggested Problems: #2, 9, 12, 13, 18, 21, 27, 31, 34.
     
  2. The composition of two functions consists of a function within another function y = f(g(x)).
     
  3. The chain rule enables you to take the derivative of the composition of two functions.
     
  4. The chain rule: If f(x) and g(x) are differentiable functions, and if y = f(u) and u = g(x), then
     dy/dx = dy/du · du/dx.
  5. The chain rule: If f(x) and g(x) are differentiable functions, and if y = f(g(x)), then
     dy/dx = f(g(x)) · g(x).
  6. The chain rule applied to familiar functions: If u is a differentiable function of x, then
     d/dx ( uⁿ ) = n u^n-1 · du/dx. d/dx ( e^u ) = e^u · du/dx. d/dx ( a^u ) = ln(a) · a^u · du/dx. d/dx ( ln(u) ) = ( 1/u ) · du/dx.
  
  
  
• SECTION 4.4: The product and quotient rules
  
  
  1. Suggested Problems: #3, 6, 7, 9, 12, 18, 22, 24, 25, 26, 28.
     
  2. The product rule: If f(x) and g(x) are differentiable functions, then
     d/dx ( f(x) · g(x) ) = f(x) g(x) + g(x) f(x).
  3. The quotient rule: If f(x) and g(x) are differentiable functions, then
     d/dx ( f(x)/g(x) ) = ( g(x) f(x) - f(x) g(x) )/( (g(x))² ), provided g(x) 0.
  
  
  

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Last modified: Fri Apr 12 2002