Derivative Formulas with Worked Numerical Examples

Distinguish a derivative function from its value at a point

For f(x)=x², the derivative function is f′(x)=2 x. At x=3, its numerical value is 6, the slope of the tangent there. A derivative formula describes how that slope varies with x; a point estimate gives only one value. Entering an expression and a point in the local form asks for the latter. It cannot replace a symbolic formula when a problem asks for f′(x), an antiderivative or an explanation of the differentiation rule.

Apply the power and constant rules

For a polynomial term ax^n, differentiation gives a n x^(n−1), and a constant differentiates to zero. Take f(x)=3 x³−5 x+2. Its derivative is 9 x²−5 because the cubic term contributes 9 x², the linear term contributes −5 and the constant contributes zero. At x=2, f′(2)=36−5=31. Enter 3*x^3-5*x+2 and point 2 in the local form to compare its numerical estimate with this exact result.

Use the product rule without dropping a term

If f(x)=u(x)v(x), then f′(x)=u′(x)v(x)+u(x)v′(x). For f(x)=(x²+1)(x³−2), the derivative is 2 x(x³−2)+3 x²(x²+1). At x=1, this equals−2+6=4. Differentiating both factors and multiplying only 2 x by 3 x² would omit the contributions from the original factors and give the wrong derivative. Expanding the polynomial first and differentiating term by term provides an independent algebraic check.

Apply the quotient rule and preserve its domain

For f(x)=(x²+1)/(x+2), the derivative is [2 x(x+2)−(x²+1)]/(x+2)², which simplifies to(x²+4 x−1)/(x+2)². The original function excludes x=−2, and the derivative is also not defined there. At x=0, the slope is −1/4. Keep the entire denominator squared and keep the subtraction order in the numerator; reversing those terms changes the sign of the result.

Track the inner derivative in a composition

For f(x)=(2 x−1)^4, the chain rule gives 4(2 x−1)^3 multiplied by the inner derivative 2, so f′(x)=8(2 x−1)^3. At x=1, the derivative is 8. The factor 2 is essential because the inner expression changes twice as fast as x. This same idea gives the derivative of sin(3 x) as 3 cos(3 x), with radian arguments. A power-rule result that treats 2 x−1 as an unchanged variable misses the inner rate of change.

Check a trigonometric derivative in radians

The derivative of sin(x) is cos(x) when x is measured in radians. At zero the expected slope is 1. Enter sin(x) and point 0 for the second local example. A degree-based sine function would need an additional π/180 factor, so the angle convention is not an optional display detail in differentiation. The expression evaluator uses radian trigonometric functions; do not substitute degree numbers into a calculus formula without making the corresponding conversion.

Understand the finite-difference method

The local estimate uses[f(x+h)−f(x−h)]/(2 h), with h=0.000001 times the larger of 1 and |x|. This balances a small step with the numerical scale, but it is still an approximation without a universal error guarantee. Cancellation, rapid curvature, oscillation and limited floating-point resolution can distort the estimate. A disagreement between one-step slopes is a reason to investigate, not a proof that a derivative does not exist. Analytic differentiation is particularly valuable when the sample step is poorly matched to the function.

Check smoothness before interpreting a slope

The function |x| has left slope−1 and right slope 1 at zero, so it has no ordinary derivative there. A central symmetric difference alone would produce zero and conceal the corner. Conversely, a steep but smooth polynomial can have very different slopes at a finite distance on either side while still having a derivative at the centre. Use the function's definition and one-sided limiting slopes to settle such cases, and report the local output as a numerical check of a smooth-point example rather than a symbolic theorem.

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