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Differentiate a polynomial using the power rule.
Key values: f(x) = x^3 - 2x^2 + x · 1st derivative
Differentiate a composite function with sin(x^2).
Key values: f(x) = sin(x^2) · Chain rule applied
Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.
Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0
Compute the second derivative to analyze concavity.
Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative
Enter a mathematical expression and click Calculate to see the symbolic derivative with step-by-step solution and interactive graph.
When differentiating the product of two functions, you cannot simply multiply the derivatives. Instead:
A useful mnemonic: “derivative of the first times the second, plus the first times the derivative of the second.”
| Function | f(x) | g(x) | Result |
|---|---|---|---|
The quotient rule is derived from the product rule applied to :
Mnemonic (lo-di-hi): “Low d-high minus high d-low, over the square of what's below.” Here “high” is the numerator and “low” is the denominator.
For three functions, the pattern extends symmetrically:
In general, for factors, there are terms, each differentiating one factor and leaving the rest unchanged.
For products of many factors or functions like , take the logarithm first:
This converts products into sums, making complex products manageable.
The product rule states . You cannot simply multiply the derivatives. A useful mnemonic: “derivative of the first times the second, plus the first times the derivative of the second.”
For three factors: . In general, for factors there are terms, each differentiating exactly one factor and leaving the rest unchanged.
The quotient rule is . It is derived from applying the product rule to . The mnemonic “lo d-high minus high d-low, over the square of what's below” helps remember the formula.
Use logarithmic differentiation when you have products of many factors or expressions like . Taking the log converts products to sums: , making differentiation simpler.
Sometimes. For products of polynomials like , expanding first and using the power rule is simpler. But for products involving trig, exponential, or logarithmic functions (like ), the product rule is unavoidable.