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Explicit Differentiation

Explicit Differentiation

The process of finding the derivative of an explicit function. For example, the explicit function y = x2 – 7x + 1 has derivative y' = 2x – 7.

 

 

See also

Implicit differentiation

Worked Example

Problem: Find the derivative of y = 3x⁴ − 5x² + 2x using explicit differentiation.
Step 1: Apply the power rule term by term. The power rule states that the derivative of xⁿ is nxⁿ⁻¹.
ddx(3x4)=12x3\frac{d}{dx}(3x^4) = 12x^3
Step 2: Differentiate the second term.
ddx(−5x2)=−10x\frac{d}{dx}(-5x^2) = -10x
Step 3: Differentiate the third term.
ddx(2x)=2\frac{d}{dx}(2x) = 2
Step 4: Combine all the results.
y′=12x3−10x+2y' = 12x^3 - 10x + 2
Answer: y' = 12x³ − 10x + 2

Why It Matters

Explicit differentiation is the default method you use whenever a function is solved for y. It forms the foundation for all derivative calculations in calculus. Understanding it clearly is essential before moving on to implicit differentiation, where y cannot be easily isolated.

Common Mistakes

Mistake: Confusing explicit differentiation with implicit differentiation and unnecessarily using dy/dx on both sides of an equation already solved for y.
Correction: If y is already isolated as a function of x (e.g., y = x³ + 2x), you can differentiate the right side directly using standard rules — no need for implicit techniques.

Related Terms