What a derivative tells you
The derivative f′(x) is the slope of the graph of f at each point: how fast f is changing right there. Use it for velocity from position, marginal cost from total cost, the tangent line to a curve, or to find where a function rises, falls and turns.
How it's calculated
The calculator reads your function into a tree of pieces (sums, products, powers and functions) and works from the outside in, applying one rule at each level. Then it tidies the result so you don't see clutter like 1·x⁰.
Product rule: (fg)′ = f′g + fg′
Quotient rule: (f/g)′ = (f′g − fg′)/g²
Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x)
The trig rules are d/dx sin x = cos x, d/dx cos x = −sin x, d/dx tan x = sec²x, d/dx sec x = sec x tan x, d/dx csc x = −csc x cot x and d/dx cot x = −csc²x.
Worked example
Differentiate f(x) = x² sin x and find the slope at x = π/2.
- It's a product, so use the product rule with f = x² and g = sin x.
- f′ = 2x by the power rule, and g′ = cos x.
- So f′(x) = 2x·sin x + x²·cos x.
- At x = π/2, sin x = 1 and cos x = 0, so f′(π/2) = 2(π/2)(1) + 0 = π ≈ 3.14159.
A chain rule example: d/dx arcsin(2x) = 2/√(1 − 4x²). The outer function is arcsin u with derivative 1/√(1 − u²), and the inside u = 2x has derivative 2.
Derivatives of inverse trig functions
| Function | Derivative | Where it holds |
|---|---|---|
| arcsin x | 1/√(1 − x²) | −1 < x < 1 |
| arccos x | −1/√(1 − x²) | −1 < x < 1 |
| arctan x | 1/(1 + x²) | all x |
| arccot x | −1/(1 + x²) | all x |
| arcsec x | 1/(|x|√(x² − 1)) | |x| > 1 |
| arccsc x | −1/(|x|√(x² − 1)) | |x| > 1 |
With the chain rule, replace x by u and multiply by u′. For example, d/dx arctan(x²) = 2x/(1 + x⁴).
Exponential and logarithm rules
| Function | Derivative |
|---|---|
| eˣ | eˣ |
| e^g(x) | e^g(x) · g′(x) |
| aˣ (a > 0) | aˣ · ln a |
| ln x (x > 0) | 1/x |
| ln g(x) | g′(x)/g(x) |
| log_b x | 1/(x · ln b) |
Logarithmic differentiation
When both the base and the power contain x, as in y = xˣ, none of the rules above fits. Take ln of both sides, ln y = x ln x, differentiate (y′/y = ln x + 1) and multiply by y. So d/dx xˣ = xˣ(ln x + 1). The calculator does this automatically and shows the steps.
Limitations
- One variable, x. Partial derivatives of f(x, y) are on the directional derivative calculator.
- The answer is simplified sensibly, not into one "canonical" form, so it may look different from your textbook while being equal. Check by evaluating both at a point.
- Simplifying can hide where f is undefined. For example, x/x simplifies to 1, but the original isn't defined at 0.
- Implicit differentiation (an equation in x and y) isn't supported.
Sources
- OpenStax Calculus Volume 1, 3.3: Differentiation rules (power, product and quotient rules)
- OpenStax Calculus Volume 1, 3.6: The chain rule
- OpenStax Calculus Volume 1, 3.5: Derivatives of trigonometric functions
- OpenStax Calculus Volume 1, 3.7: Derivatives of inverse functions (Theorem 3.13, inverse trig derivatives)
- OpenStax Calculus Volume 1, 3.9: Derivatives of exponential and logarithmic functions
- Paul's Online Notes (Lamar University): Logarithmic differentiation
Learn: Derivative rules table