Derivative of -x
Answer
Derivative of -x
- Rule Used
- Constant Multiple Rule
- Steps
- -x 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: -1
d/dx -x = -1. The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
Instant step-by-step solution for -x
Derivative of -x = -1 Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-minus-x
Derivative
-x → -1engine: -1
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
−x is the line through the origin with negative slope. There is no extra constant to drop and no power higher than 1, so the derivative is exactly the coefficient −1. It is the same answer as the derivative of 1−x, because those two lines are vertical shifts of each other.
Apply the power rule to -x. The form to remember, already simplified, is -1.
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 -1. That number is the slope of -x there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = -x.
- Apply the power rule to -x. The form to remember, already simplified, is -1.
- At x = 2, substitute into the simplified form -1. That number is the slope of -x there; a secant over a small step in x lands next to it.
Common mistakes
Answering 1
Answering 1 loses the sign. The line y = −x falls at 45 degrees, so every secant slope is −1.
Keeping x in the derivative of -x
-x is a line, so the derivative is a constant slope. An answer that still contains x is the original function, not its slope.
Related rule and examples
- Power rule — The rule page lists this example and the others in the same family.
- Derivative of 2x — Same family of differentiation, different function.
- Derivative of 5x — Same family of differentiation, different function.
- Derivative of x+1 — Same family of differentiation, different function.
- Derivative of 1-x — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of -x?
-1. −x is the line through the origin with negative slope. There is no extra constant to drop and no power higher than 1, so the derivative is exactly the coefficient −1. It is the same answer as the derivative of 1−x, because those two lines are vertical shifts of each other.
Which wrong answer shows up for -x?
Answering 1 loses the sign. The line y = −x falls at 45 degrees, so every secant slope is −1.
Where is the derivative of -x defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.