Calculus Volume 1 · free preview
Differentiation Rules
The power, sum, product, quotient, and constant rules turn differentiation from a limit computation into an algebraic routine applicable to any elementary combination.
1. Constant, power, sum, and multiple rules
Constant, power, sum, and multiple rules
2. The product rule
The product rule
3. The quotient rule
The quotient rule
4. Combining the rules
Combining the rules
5. Power rule at a fractional exponent
What is the derivative of f(x) = x^(1/2) = sqrt(x) for positive x?
6. Product rule application
What is the derivative of f(x) = x sin x?
7. Quotient rule sign
Differentiating f(x) = u/v by the quotient rule, which numerator is correct?
8. Derivative of a constant multiple
What is the derivative of the function f(x) = 7x^3 with respect to x?
9. Stacking the rules
To differentiate f(x) = (x^2 + 1) sin x, which sequence of rule applications is correct?
10. Rule for a sum
What is the derivative of the sum f(x) = x^2 + 3x with respect to x?
11. Derivative of x to the fifth
By the power rule, what is the derivative of x^5? Answer like 5x^4.
12. Derivative of a constant
The constant function f(x) = 9 has a horizontal graph. What is its derivative? Answer with a number.
13. Quotient at integers
What is the derivative of f(x) = 1/x at x = 2? Answer as a fraction like -1/4.
14. Sum rule count
The polynomial p(x) = x^3 + 2x^2 - x + 4 has four terms. How many of those terms survive differentiation with a changed coefficient? Answer with a number.
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