Differentiation
Differentiation transforms how students understand rates of change, moving from average velocity calculations to instantaneous motion analysis. When teaching the power rule d/dx[x³] = 3x², students grasp that derivatives reveal the slope at any point on a curve.
Try it right now
Why it matters
Differentiation drives real-world problem solving across engineering, economics, and physics. Engineers use derivatives to optimize bridge designs, calculating maximum stress points where the derivative equals zero. Stock market analysts apply differentiation to identify price trend reversals, with derivatives indicating when growth rates change from positive to negative. In physics, velocity is the derivative of position, and acceleration is the derivative of velocity—making d/dx essential for understanding motion. Medical researchers use derivatives to model drug concentration changes in blood, determining optimal dosing schedules. The R1 and R2 curriculum standards emphasize these applications, while CCSS.HSF.IF connects derivatives to function interpretation and analysis.
How to solve differentiation
Differentiation
- Power rule: d/dx [xⁿ] = nxⁿ⁻¹.
- Chain rule: d/dx [f(g(x))] = f'(g(x)) × g'(x).
- Product rule: d/dx [uv] = u'v + uv'.
- Derivative = gradient of the tangent = instantaneous rate of change.
Example: f(x) = 3x⁴ → f'(x) = 12x³. At x=2: f'(2) = 96.
Worked examples
Differentiate: f(x) = 2 x2
Answer: f'(x) = 4 x
- Apply the power rule: d/dx[ax^n] = nax^(n-1) → f'(x) = 2·2x^1 = 4 x — Multiply the exponent 2 by the coefficient 2, then reduce the exponent by 1.
Differentiate: f(x) = 2 x3 + x2 - 6 x + 3
Answer: f'(x) = 6 x2 + 2 x - 6
- Write out the rule → d/dx[x^n] = n·x^(n-1) — The power rule: multiply by the exponent, then reduce the exponent by 1.
- Differentiate 2 x^3 → 3·2x^2 = 6 x^2 — Exponent 3 comes down as a factor, exponent becomes 3−1 = 2.
- Differentiate x^2 → 2·1x = 2 x — Exponent 2 comes down, exponent becomes 2−1 = 1.
- Differentiate -6x → -6 — The derivative of kx is just k. The constant d vanishes.
- Combine all terms → f'(x) = 6 x^2 + 2 x - 6 — Write the derivative as one expression.
Differentiate: f(x) = 3 sin(x)
Answer: f'(x) = 3 cos(x)
- Apply the rule: d/dx[sin(x)] = cos(x) → f'(x) = 3 cos(x) — The constant 3 is carried through.
Common mistakes
- ✗Students often forget to reduce the exponent when applying the power rule, writing d/dx[x³] = 3x³ instead of 3x².
- ✗When differentiating 2x⁴ + 3x, students frequently write 8x⁴ + 3 instead of 8x³ + 3, forgetting the power rule completely.
- ✗Chain rule errors occur when students differentiate (2x+1)³ as 6x² instead of 3(2x+1)² × 2 = 6(2x+1)².
- ✗Students incorrectly differentiate sin(3x) as 3sin(x) instead of 3cos(3x), missing the chain rule coefficient of 3.
Practice on your own
Generate unlimited differentiation practice problems with step-by-step solutions using MathAnvil's free worksheet generator.
Generate free worksheets →