Derivative of x*e^x
Answer
Derivative of x*e^x
- Rule Used
- Product Rule
- Steps
- Identify a product of two functions of x in x*e^x: use the product rule (uv)' = u'v + uv'. Each factor is differentiated once while the other is held as-is, then the two terms are added.
- Differentiate each factor and combine.
- Final answer: e^x+e^x*x
d/dx x e^x = e^x+e^x*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 x*e^x
Derivative of x*e^x = e^x+e^x*x Solve step-by-step for free: https://derivativecalculatorai.com/derivative-of-x-e-to-the-x
Derivative
x e^x → e^x+e^x*xengine: e^x+e^x*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
x e^x is the standard product of x and the exponential. Because e^x is its own derivative, both terms contain e^x: one from differentiating x, and one from differentiating the exponential. Factor e^x(x + 1). Answering x e^x again means only one factor was differentiated.
Apply the product rule to x e^x. The form to remember, already simplified, is e^x+e^x*x.
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 e^x+e^x*x. That number is the slope of x e^x there; a secant over a small step in x lands next to it.
Worked derivative
- Write f(x) = x e^x.
- Apply the product rule to x e^x. The form to remember, already simplified, is e^x+e^x*x.
- At x = 0, substitute into the simplified form e^x+e^x*x. That number is the slope of x e^x there; a secant over a small step in x lands next to it.
Common mistakes
Answering x*e^x
x e^x is the original function. The product rule adds a second copy of e^x, the one that comes from the derivative of x.
Differentiating only one factor of x e^x
Both factors depend on x. Differentiating one and copying the other, then stopping, drops a whole term.
Related rule and examples
- Product rule — The rule page lists this example and the others in the same family.
- Exponential derivatives — The rule page lists this example and the others in the same family.
- Derivative of x sin(x) — Same family of differentiation, different function.
- Derivative of x cos(x) — Same family of differentiation, different function.
- Derivative of x ln(x) — Same family of differentiation, different function.
Frequently asked questions
What is the derivative of x e^x?
e^x+e^x*x. x e^x is the standard product of x and the exponential. Because e^x is its own derivative, both terms contain e^x: one from differentiating x, and one from differentiating the exponential. Factor e^x(x + 1). Answering x e^x again means only one factor was differentiated.
Which wrong answer shows up for x e^x?
x e^x is the original function. The product rule adds a second copy of e^x, the one that comes from the derivative of x.
Where is the derivative of x e^x defined?
The expression is built from polynomials, sine, cosine, or exponentials, so the derivative exists for every real x.