For the following exercises, find for each function.
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Calculus Volume 1 · 3.3 Differentiation Rules
Derivatives: practice problems on the differentiation rules
24 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.
How to approach these problems
Before you differentiate, simplify. Multiply out a product of polynomials and split a fraction with a one-term denominator into separate powers of x: then the power rule, (xⁿ)′ = nxⁿ⁻¹, does everything, negative exponents included. Keep the product rule, (fg)′ = f′g + fg′, for products you can't expand, and the quotient rule, (f/g)′ = (f′g − fg′)/g², for real fractions; the order in its numerator matters. For a tangent line at x = a, find the point f(a) and the slope f′(a), then write y − f(a) = f′(a)(x − a). The usual mistakes: taking the derivative of a quotient as f′/g′, losing a sign in the quotient rule, and forgetting that a constant's derivative is zero.
Exercises 109
Exercises 111
Exercises 113
Exercises 115
Exercises 117
Exercises 119
Exercises 121
Exercises 123
Exercises 125
Exercises 127
Exercises 129
Exercises 131
Exercises 133
Exercises 135
Exercises 137
Exercises 139
Exercises 141
Exercises 143
Exercises 145
Exercises 147
Exercises 149
Review Exercises 381
Review Exercises 383
Key terms · 7
- constant multiple rule
- the derivative of a constant c multiplied by a function f is the same as the constant multiplied by the derivative: $\frac{d}{d x} (c f (x)) = c f' (x)$
- constant rule
- the derivative of a constant function is zero: $\frac{d}{d x} (c) = 0,$ where c is a constant
- difference rule
- the derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g: $\frac{d}{d x} (f (x) - g (x)) = f' (x) - g' (x)$
- power rule
- the derivative of a power function is a function in which the power on $x$ becomes the coefficient of the term and the power on $x$ in the derivative decreases by 1: If $n$ is an integer, then $\frac{d}{d x} x^n = n x^{n - 1}$
- product rule
- the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function: $\frac{d}{d x} (f (x) g (x)) = f' (x) g (x) + g' (x) f (x)$
- quotient rule
- the derivative of the quotient of two functions is the derivative of the first function times the second function minus the derivative of the second function times the first function, all divided by the square of the second function: $\frac{d}{d x} (\frac{f (x)}{g (x)}) = \frac{f' (x) g (x) - g' (x) f (x)}{{(g (x))}^2}$
- sum rule
- the derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g: $\frac{d}{d x} (f (x) + g (x)) = f' (x) + g' (x)$
Learn 3.3 on the whiteboard
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