Derivative Calculator
Calculate derivatives of functions with step-by-step differentiation and derivative rules.
Result
Function
Step-by-Step Solution
Basic Derivative Rules
Power Rule
d/dx [xⁿ] = n·xⁿ⁻¹
Multiply by the exponent and reduce the power by one.
Constant Multiple Rule
d/dx [c·f(x)] = c·f'(x)
Constants factor out of the derivative.
Sum / Difference Rule
d/dx [f ± g] = f' ± g'
Differentiate each term independently.
Product Rule
d/dx [f·g] = f'·g + f·g'
Derivative of first times second, plus first times derivative of second.
Quotient Rule
d/dx [f/g] = (f'·g − f·g') / g²
"Low d-high minus high d-low, over the square of what's below."
Chain Rule
d/dx [f(g(x))] = f'(g(x))·g'(x)
Derivative of the outer function evaluated at the inner, times derivative of the inner.
Common Derivatives
| Function f(x) | Derivative f'(x) | Rule |
|---|---|---|
| c (constant) | 0 | Constant |
| xⁿ | n·xⁿ⁻¹ | Power |
| eˣ | eˣ | Exponential |
| ln(x) | 1/x | Logarithmic |
| sin(x) | cos(x) | Trigonometric |
| cos(x) | −sin(x) | Trigonometric |
| tan(x) | sec²(x) | Trigonometric |
| cot(x) | −csc²(x) | Trigonometric |
| sec(x) | sec(x)·tan(x) | Trigonometric |
| csc(x) | −csc(x)·cot(x) | Trigonometric |
| √x | 1 / (2√x) | Power (n=½) |
Worked Examples
Example 1: Polynomial (Power Rule)
Find: d/dx [4x³ − 2x² + 7x − 5]
Step 1: Apply the sum/difference rule to each
term
Step 2: d/dx[4x³] = 12x² (power rule)
Step 3: d/dx[−2x²] = −4x (power rule)
Step 4: d/dx[7x] = 7, d/dx[−5] = 0
Result: 12x² − 4x + 7
Example 2: Chain Rule
Find: d/dx [sin(3x²)]
Step 1: Outer function: sin(u), inner: u =
3x²
Step 2: d/du[sin(u)] = cos(u) = cos(3x²)
Step 3: d/dx[3x²] = 6x
Result: cos(3x²) · 6x = 6x·cos(3x²)
Example 3: Product Rule
Find: d/dx [x² · eˣ]
Step 1: Let f = x², g = eˣ
Step 2: f' = 2x, g' = eˣ
Step 3: f'·g + f·g' = 2x·eˣ + x²·eˣ
Result: eˣ(2x + x²)
References
The differentiation rules and formulas used in this calculator are based on standard calculus from established educational resources:
Related Calculators
This calculator computes symbolic derivatives using standard differentiation rules. It supports polynomials, trigonometric, exponential, and logarithmic functions. Results are for educational and informational purposes. Please verify important calculations independently.