Power Rule & Derivatives

Trace how secant lines converge to the exact tangent line. The Power Rule states that $\frac{d}{dx}[x^n] = n x^{n-1}$. Master derivatives using the formal limit definition of slopes.

Interactive Sandbox

Secant to Tangent Limit Convergence

Drag the target value **c** directly on the graph's x-axis. Adjust interval step $h$ toward 0 to watch the green dashed secant line rotate into alignment with the red tangent line.

Curve: y = x²

Derivative Power Rule

d/dx (xⁿ) = n • xⁿ⁻¹
Secant Step (h) 1.00
Target Coordinate (c) 1.50
Drag the marker 'c' on the x-axis. Notice how smaller values of 'h' make the secant slope match the tangent derivative.
Algebraic Solver

Limit Definition & Power Rule Steps

Trace numerical secant slopes $\Delta y / \Delta x$ vs analytical derivatives $f'(c) = n c^{n-1}$.

Analytical Derivative Steps


Derivative Slopes Summary


Secant Slope (Δy / Δx)

m_sec = 4.00

Tangent Derivative Slope (f'(c))

m_tan = 3.00

Slope Difference (Δm)

Difference = 1.00

Evaluation

Derivatives Quiz

Test your knowledge of the power rule, fractional indices, negative exponents, and tangent limits.

Question 1 of 5 Score: 0/0

Evaluate the derivative of f(x) = x⁴ using the Power Rule.

Correct Answer!

Explanation text...

Power Derivative Rules


  • Standard Power: d/dx (x^n) = n x^(n-1)
  • Radical: d/dx (x^0.5) = 0.5 x^(-0.5)
  • Reciprocal: d/dx (1/x) = -1/x²