Learning Objectives
Explain why continuity does not guarantee differentiability.
Apply constant, power, sum, and constant-multiple rules.
Differentiate basic trigonometric, exponential, and logarithmic functions with their domain conditions.
Prerequisites: the derivative definition, continuity, exponent rules, and radian trigonometry. Rules shorten repeated limit calculations, but their conditions still matter.
1. A Finite Derivative Requires Matching Local Slopes
If f is differentiable at an interior point a, then f is continuous there. The converse is false: a graph can be continuous but have no single finite tangent slope. Corners, cusps, and vertical tangents are common places to check carefully.
Worked Example: A Continuous Corner
For f(x) = |x| at x = 0, the difference quotient is |h|/h.
For h > 0 it equals 1; for h < 0 it equals −1. The one-sided derivative limits disagree, so f′(0) does not exist.
However, f is continuous at zero because its nearby values approach f(0) = 0. Continuity alone is not enough.
2. Differentiate Powers and Sums
Function | Derivative |
|---|---|
Constant c | 0 |
xⁿ | n x^(n − 1), wherever the rule applies |
c f(x) | c f′(x) |
f(x) + g(x) | f′(x) + g′(x) |
For positive integer powers, the power rule holds for every real x. Negative integer powers exclude x = 0. For general real powers, x > 0 gives a standard real domain on which the rule applies; endpoints and any extensions to negative inputs need separate checks.
Worked Example: Rewrite before Differentiating
Let f(x) = 3x⁴ − 2/x + √x, with x > 0. Rewrite it as 3x⁴ − 2x⁻¹ + x^(1/2).
Apply the power rule term by term: f′(x) = 12x³ + 2x⁻² + (1/2)x^(−1/2).
Equivalently, f′(x) = 12x³ + 2/x² + 1/(2√x). The stated domain avoids the zero denominators.
3. Standard Functions Have Standard Derivatives
Function | Derivative | Condition |
|---|---|---|
sin x | cos x | Radians |
cos x | −sin x | Radians |
tan x | sec²x | Radians; cos x ≠ 0 |
eˣ | eˣ | All real x |
aˣ | aˣ ln a | a > 0 |
ln x | 1/x | x > 0 |
The trigonometric derivatives depend on radian measure. The exponential function eˣ is its own derivative; this property does not hold unchanged for every base. A constant base a introduces the factor ln a.
Worked Example: Combine Known Derivatives
For g(x) = 4 sin x − 3 cos x + 2eˣ + ln x on x > 0, differentiate each term.
The result is g′(x) = 4 cos x + 3 sin x + 2eˣ + 1/x.
The plus sign on 3 sin x comes from multiplying the original −3 by the negative derivative of cos x.
4. A Piecewise Join Needs More than Equal Heights
At a piecewise boundary, first check continuity. Then compare the limiting slopes from both sides using the formulas on their respective sides. A jump rules out differentiability immediately. Matching heights but different slopes produces a corner.
Common Mistakes
The derivative of a constant is zero, not the constant itself.
The power rule lowers the exponent by one and multiplies by the original exponent.
These sum rules do not say that the derivative of a product is the product of derivatives.
5. Practice
1. Differentiate f(x) = 5x³ − 4x + 9.
2. Differentiate g(x) = x⁻² + 3√x for x > 0.
3. Differentiate h(x) = 2 cos x + 5ˣ.
4. Can a function with a jump discontinuity at x = 2 be differentiable there? Explain.
Worked Solutions
1. f′(x) = 15x² − 4. The derivative of 9 is zero.
2. g′(x) = −2x⁻³ + (3/2)x^(−1/2).
3. h′(x) = −2 sin x + 5ˣ ln 5.
4. No. Differentiability implies continuity, and a jump violates continuity.
6. Summary
- Differentiability implies continuity, but not conversely.
- Apply basic derivative rules term by term when operations are sums.
- Keep domain restrictions and radian measure visible.
- Check both values and slopes at a piecewise boundary.