Concepts: derivatives from first principles and derivative rules

5.1 The derivative from the definition

The derivative at a point is the limit of the difference quotient. To compute it from first principles, write f(a plus h) minus f(a), expand or simplify, divide by h and let h tend to zero. The cancellation of h is the essential step; a question that cannot cancel h is usually testing the standard limits rather than algebra.

For a polynomial the expansion is straightforward and the limit is easy. For sine and cosine the difference quotient leads to the two standard trigonometric limits, which is why those limits are proved before derivatives are introduced. The first-principles questions in the exercise set are mostly polynomials, rational functions and the sine function, so the algebra stays manageable.

A derivative at a point is a number, while the derivative function assigns a number to every point. The phrase 'find the derivative' usually means the function, and 'find f prime at 2' means the number. Reading the question carefully avoids writing a function where a number is wanted.

5.2 Algebra of derivatives and the product rule

The derivative of a sum is the sum of the derivatives, and a constant multiple can be pulled out. These two rules reduce any polynomial to the power rule applied term by term. For a product, the product rule applies: the derivative of the first times the second plus the first times the derivative of the second. The order does not matter for the final sum, but writing both terms avoids missing one.

The derivative of a rational function can be computed with the quotient rule or by rewriting it as a sum of powers. Rewriting is usually faster in this chapter because negative powers follow the same power rule. A constant has derivative zero, and any constant multiple of a function has the constant times the derivative.

The derivative of a polynomial is again a polynomial of one lower degree, so the degree drops by one with each differentiation. This is a quick check on the answer: differentiating a cubic must give a quadratic, and differentiating a quadratic must give a linear function.

5.3 Derivatives of trigonometric functions

The derivative of sine is cosine and the derivative of cosine is minus sine. The derivative of tangent is secant squared, obtained by writing tangent as sine over cosine and applying the quotient rule. The derivatives of cotangent, secant and cosecant follow the same route, and they are listed in the formula sheet with their signs.

Differentiating combinations of trigonometric functions uses the algebra rules. A product such as x sine x needs the product rule and gives sine x plus x cosine x. A constant multiple of a sum is differentiated term by term. A quotient of trigonometric functions is handled by the quotient rule or by rewriting in terms of sine and cosine.

The signs are the main source of error. The derivative of cosine is negative, the derivative of cotangent is negative, and the derivative of cosecant is negative; the other three are positive. Grouping the negative ones in the memory line 'cos, cot, cosec' is a reliable way to avoid swapping a sign.

5.4 Exercise 12.2 practice

Exercise 12.2 asks for derivatives from first principles, derivatives by the rules, and derivatives at stated points. A typical first-principles question is the derivative of x squared plus 3x at x equal to 2: the difference quotient simplifies to 4 plus 3 plus h, and the limit is 7, which also matches the rule value 2x plus 3 at x equal to 2.

Another item asks for the derivative of a polynomial such as 3x to the fourth minus 2x cubed plus x minus 5. Applying the power rule term by term gives 12x cubed minus 6x squared plus 1. Checking the degree drop from four to three confirms the answer is plausible.

Trigonometric items include the derivative of sine x plus cosine x, which is cosine x minus sine x, and the derivative of x cosine x, which is cosine x minus x sine x by the product rule. The exercise also includes derivatives at a point, where the derivative function is found first and the point is substituted afterwards; substituting before differentiating is a common error and always gives zero.

A final check for derivative answers is dimensional consistency. Differentiating a polynomial reduces its degree by one; differentiating a product of a polynomial and a trigonometric function keeps both kinds of term; differentiating a constant gives zero. If the answer has the wrong shape, a term has been dropped, and the product rule is the usual place to look.