Derivative of 3x^2
Answer
Derivative of 3x^2
- Rule Used
- Constant Multiple Rule
- Steps
- is a constant times a function of x. The constant is a fixed stretch of the slope, so it factors out and the remaining function is differentiated.
- Final answer:
d/dx 3x^2 = 6*x. The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
Instant step-by-step solution for 3x^2
Derivative of 3x^2 = 6*x Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-3x-squared
Derivative
3x^2 → 6*xengine: 6*x
Graph of the function and its derivative
The solid curve is f. The dashed curve is f′. Vertical gaps are asymptotes or jumps, not extra zeros.
Why the derivative has this form
The 3 in front of x^2 is a constant stretch, so it survives differentiation, and the power rule still turns x^2 into 2x. The two numbers multiply: 3·2x = 6x. Nothing in the constant is differentiated, because a constant has slope zero and only scales the slope of x^2.
Apply the power rule to 3x^2. The form to remember, already simplified, is 6*x.
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
At x = 2, substitute into the simplified form 6*x. That number is the slope of 3x^2 there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = 3x^2.
- Apply the power rule to 3x^2. The form to remember, already simplified, is 6*x.
- At x = 2, substitute into the simplified form 6*x. That number is the slope of 3x^2 there; a secant over a small step in x lands next to it.
Common mistakes
Answering 3*x
3x forgets to differentiate the square. The 3 stays, but the power still contributes a 2, and 3 times 2 is 6, not 3.
Forgetting to lower the exponent of 3x^2
The front coefficient of 3x^2 is only half the work. The exponent itself has to decrease by 1. Leaving the original power in place describes a different curve.
Related rule and examples
- Power rule — The rule page lists this example and the others in the same family.
- Derivative of x^2 — Same family of differentiation, different function.
- Derivative of x^3 — Same family of differentiation, different function.
- Derivative of x^4 — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of 3x^2?
6*x. The 3 in front of x^2 is a constant stretch, so it survives differentiation, and the power rule still turns x^2 into 2x. The two numbers multiply: 3·2x = 6x. Nothing in the constant is differentiated, because a constant has slope zero and only scales the slope of x^2.
Which wrong answer shows up for 3x^2?
3x forgets to differentiate the square. The 3 stays, but the power still contributes a 2, and 3 times 2 is 6, not 3.
Where is the derivative of 3x^2 defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.