Why does the difference quotient for sin x use the sine addition formula?

The difference quotient (sin(x+h) − sin x)/h cannot be simplified without expanding sin(x+h). The sine addition formula sin(x+h) = sin x cos h + cos x sin h is the only identity that separates x and h into separate terms that can be grouped and cancelled.

What happens if h does not cancel from the numerator?

If h does not cancel algebraically from the difference quotient, the function is not differentiable at that x. Cancellation is required because the denominator h becomes zero in the limit, and only a factor of h in the numerator prevents division by zero.

Can I use the symmetric difference quotient instead for sin x?

The symmetric difference quotient (sin(x+h) − sin(x−h))/(2h) approximates the derivative numerically but does not match the limit definition in Stewart or OpenStax. For a formal derivation from first principles, use the standard difference quotient with a single increment h.

Where do the limits sin h/h → 1 and (cos h − 1)/h → 0 come from?

These limits are proven geometrically using the unit circle and the squeeze theorem. They are standard results in calculus textbooks; when working with the difference quotient, you apply them as known facts rather than re-proving them each time.

Difference Quotient of sin x, cos x and e^x

Difference Quotient Sin X: The First Step to the Derivative of Sine

The difference quotient for sin x is (sin(x+h) − sin x)/h. This is the expression you must evaluate to find the derivative of sin x from first principles, as laid out in Stewart Calculus sections 2.7-2.8. The algebraic machinery that makes this work is the sine addition formula: sin(x+h) = sin x cos h + cos x sin h. Substituting that identity into the numerator converts a confusing expression into a pair of limits you already know. You cannot simplify sin(x+h) − sin x any other way; the identity is the only path forward.

Deriving the Difference Quotient for Sin X

Start with the difference quotient formula: (f(x+h) − f(x))/h. For f(x) = sin x, this becomes (sin(x+h) − sin x)/h. Replace sin(x+h) using the identity sin(A+B) = sin A cos B + cos A sin B. You get:

(sin x cos h + cos x sin h − sin x)/h.

Group the sin x terms together: sin x (cos h − 1) + cos x sin h, all over h. Split the fraction into two separate quotients:

sin x · (cos h − 1)/h + cos x · (sin h)/h.

This is the difference quotient for sin x fully factored. At this point you have not taken a limit; you have only algebraically rearranged the secant line slope into a form that contains the two fundamental trig limits. The next step is to let h approach zero.

The Two Limits That Make It Work: Sin H / H and (Cos H − 1) / H

The difference quotient for sin x simplifies to sin x · (cos h − 1)/h + cos x · (sin h)/h. To take the limit as h→0, you need two specific results:

lim_{h→0} (sin h)/h = 1. This limit is derived geometrically in calculus textbooks and is the reason the derivative of sin x is cos x. The proof uses the squeeze theorem on the unit circle; you take it as given when working with the difference quotient.

lim_{h→0} (cos h − 1)/h = 0. This limit follows from the identity cos h − 1 = −2 sin²(h/2) and the previous limit. It is the term that disappears, leaving only the cos x term.

Apply these limits to the factored difference quotient. The term sin x · (cos h − 1)/h becomes sin x · 0 = 0. The term cos x · (sin h)/h becomes cos x · 1 = cos x. The limit of the difference quotient as h→0 is therefore cos x. That is the derivative of sin x from first principles.

Difference Quotient Cos X: Using the Cosine Addition Formula

The difference quotient for cos x follows the same pattern. Write (cos(x+h) − cos x)/h. Use the identity cos(A+B) = cos A cos B − sin A sin B. Substitute: (cos x cos h − sin x sin h − cos x)/h. Group the cos x terms: cos x (cos h − 1) − sin x sin h, all over h. Split into two fractions:

cos x · (cos h − 1)/h − sin x · (sin h)/h.

Take the limit as h→0. The first term becomes cos x · 0 = 0. The second term becomes −sin x · 1 = −sin x. The derivative of cos x is −sin x. The algebra is nearly identical to the sin x derivation; the only difference is the minus sign from the cosine addition formula. OpenStax Calculus Volume 1 section 3.1 uses exactly this derivation to define the derivative from the difference quotient.

Difference Quotient E^X: The Exponential That Stays Itself

The Factoring Step

The difference quotient for e^x is (e^(x+h) − e^x)/h. Factor out e^x: e^x(e^h − 1)/h. The limit as h→0 of (e^h − 1)/h is 1. This is a defining property of the natural exponential function. The derivative of e^x is e^x. This derivation is simpler than the trig versions because no addition formula is required; the factoring step is immediate. Stewart Calculus section 1.4 covers exponential functions before section 2.7 introduces the difference quotient, so the e^x derivation is a natural bridge between the two chapters.

General Exponential B^X

For a general exponential b^x, use the identity b^x = e^(x ln b). The difference quotient becomes (b^(x+h) − b^x)/h = b^x(b^h − 1)/h. The limit lim_{h→0} (b^h − 1)/h = ln b. The derivative of b^x is b^x ln b. This general case covers all bases, including 2^x, 10^x, and any other exponential students encounter in OpenStax Precalculus section 1.3.

Numeric Check With a Calculator: The H-Table

Before trusting the algebra, run a numeric check. Pick a specific x, say x = π/3 (60°). Let h be a small number like 0.01. Compute the difference quotient (sin(π/3 + 0.01) − sin(π/3))/0.01. A standard calculator gives roughly 0.5. The derivative cos(π/3) is 0.5. The numeric value is close, and it gets closer as h shrinks: try h = 0.001, then 0.0001. Build a table of h values and watch the quotient approach 0.5. This confirms that the limit you derived algebraically matches the actual behavior of the secant line slopes. The most common failure here is using a numeric h and stopping without taking the limit; the algebraic cancellation must work for any h, not just the one you typed into the calculator.

The difference quotient sin x and trig functions