Derivative of 5x
Answer
Derivative of 5x
- 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: 5
d/dx 5x = 5. The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
Instant step-by-step solution for 5x
Derivative of 5x = 5 Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-5x
Derivative
5x → 5engine: 5
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
5x is the same kind of line as 2x, but the slope is 5 rather than 2. The graphs are parallel only if the coefficients match; here they do not. The derivative has to keep the 5 and drop the x, because d/dx(x) = 1 and a constant multiple factors out.
Apply the power rule to 5x. The form to remember, already simplified, is 5.
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.
At x = 1, substitute into the simplified form 5. That number is the slope of 5x there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = 5x.
- Apply the power rule to 5x. The form to remember, already simplified, is 5.
- At x = 1, substitute into the simplified form 5. That number is the slope of 5x there; a secant over a small step in x lands next to it.
Common mistakes
Answering 5*x
Answering 2, or any slope borrowed from a different line, ignores the coefficient that is actually written in front of x.
Keeping x in the derivative of 5x
5x 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 x+1 — Same family of differentiation, different function.
- Derivative of 1-x — Same family of differentiation, different function.
- Derivative of -x — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of 5x?
5. 5x is the same kind of line as 2x, but the slope is 5 rather than 2. The graphs are parallel only if the coefficients match; here they do not. The derivative has to keep the 5 and drop the x, because d/dx(x) = 1 and a constant multiple factors out.
Which wrong answer shows up for 5x?
Answering 2, or any slope borrowed from a different line, ignores the coefficient that is actually written in front of x.
Where is the derivative of 5x defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.