Derivative of 1-x
Answer
Derivative of 1-x
- Rule Used
- Sum Rule
- Steps
- Differentiate 1-x term by term. The sum rule says the derivative of a sum is the sum of the derivatives, and the derivative of a constant term is 0, so that term disappears.
- Final answer: -1
d/dx 1-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 1-x
Derivative of 1-x = -1 Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-1-minus-x
Derivative
1-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
1-x is a line that falls as x grows. Rewrite it as -x + 1: the slope is the coefficient of x, which is -1, and the constant 1 still contributes nothing. The negative sign is the whole derivative; dropping it describes a line that climbs instead of descending.
Apply the power rule to 1-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 = 0, substitute into the simplified form -1. That number is the slope of 1-x there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = 1-x.
- Apply the power rule to 1-x. The form to remember, already simplified, is -1.
- At x = 0, substitute into the simplified form -1. That number is the slope of 1-x there; a secant over a small step in x lands next to it.
Common mistakes
Answering 1
Answering 1 keeps the constant and loses the minus on x. The graph decreases through the intercept, so the slope cannot be positive.
Keeping x in the derivative of 1-x
1-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 -x — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of 1-x?
-1. 1-x is a line that falls as x grows. Rewrite it as -x + 1: the slope is the coefficient of x, which is -1, and the constant 1 still contributes nothing. The negative sign is the whole derivative; dropping it describes a line that climbs instead of descending.
Which wrong answer shows up for 1-x?
Answering 1 keeps the constant and loses the minus on x. The graph decreases through the intercept, so the slope cannot be positive.
Where is the derivative of 1-x defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.