Differentiation Calculator
Find the derivative of any function instantly — with full step-by-step working. Supports power, product, quotient and chain rules, plus trig, exponential and logarithmic functions.
Differentiation Calculator
Enter a function in terms of x to differentiate
Use ^ for powers, * for multiply, sqrt(x), sin(x), ln(x), e^x
First derivative is dy/dx; second is d²y/dx²; third is d³y/dx³
Optional: enter a value of x to calculate a numerical result for f′(x)
Derivative Result
Step-by-step differentiation working
Enter a function above and click Differentiate to see the derivative with full step-by-step working and rule identification.
Differentiation Rules
The core rules used to differentiate any function. This calculator identifies and applies the appropriate rule for each term.
d/dx [x½] = ½x⁻½
d/dx [e^f(x)] = f′(x)·e^f(x)
d/dx [ln f(x)] = f′(x)/f(x)
d/dx [cos x] = −sin x
d/dx [tan x] = sec²x
d/dx [c·f(x)] = c·f′(x)
d/dx [f±g] = f′±g′
Differentiation FAQ
Everything you need to know about finding derivatives and applying differentiation rules.
Differentiation is the process of finding the derivative of a function. The derivative measures how fast a function's output changes as its input changes. If f(x) is a function, its derivative f′(x) (also written dy/dx) gives the slope of the tangent line to the curve at any point x. Differentiation is fundamental to calculus and is used in physics, engineering, economics, and any field involving rates of change.
The power rule states: if f(x) = xⁿ, then f′(x) = n·xⁿ⁻¹. For example, the derivative of x³ is 3x², and the derivative of x⁵ is 5x⁴. The rule applies to any real exponent, including fractions and negatives — for example, the derivative of √x = x^(½) is ½x^(-½), and the derivative of 1/x = x⁻¹ is −x⁻².
The chain rule differentiates composite (nested) functions. If y = f(g(x)), then dy/dx = f′(g(x)) × g′(x). In plain words: differentiate the outer function leaving the inner unchanged, then multiply by the derivative of the inner function. For example, to differentiate sin(x²): outer = sin, inner = x². Derivative = cos(x²) × 2x = 2x·cos(x²).
The product rule applies when differentiating two functions multiplied together. If y = u(x)·v(x), then dy/dx = u′v + uv′. Differentiate the first and multiply by the second, then add the first multiplied by the derivative of the second. For example, to differentiate x²·sin(x): dy/dx = 2x·sin(x) + x²·cos(x).
The quotient rule differentiates a ratio of two functions. If y = u(x)/v(x), then dy/dx = (u′v − uv′) / v². A helpful mnemonic is "lo d-hi minus hi d-lo, over lo squared". For example, to differentiate sin(x)/x²: dy/dx = (cos(x)·x² − sin(x)·2x) / x⁴, which simplifies to (x·cos(x) − 2sin(x)) / x³.
The standard trig derivatives to memorise: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(cosec x) = −cosec x·cot x, d/dx(sec x) = sec x·tan x, d/dx(cot x) = −cosec²x. These are commonly combined with the chain rule for composite functions such as sin(3x) or cos(x²).
The derivative of eˣ is eˣ — it is the only function that equals its own derivative. The derivative of ln(x) is 1/x (for x > 0). With the chain rule: d/dx[e^f(x)] = f′(x)·e^f(x) and d/dx[ln(f(x))] = f′(x)/f(x). These rules appear constantly in growth models, physics and statistics.
The first derivative f′(x) gives the rate of change (slope) of the original function. The second derivative f″(x) is the derivative of f′(x) — it measures the concavity (curvature). If f″(x) > 0, the curve is concave up; if f″(x) < 0, it is concave down. The second derivative test classifies stationary points: f″ > 0 means local minimum, f″ < 0 means local maximum.