Derivative of e^(x²)
The exponent changes at a variable rate. Differentiate the exponential, then multiply by the derivative of x².
Step-by-step solution
Write u=x². The outer function is exp(u).
The outer derivative is exp(u); the inner derivative is 2x.
f′(x)=exp(x²)·2x=2x·exp(x²). The exponent stays x².
Domain and tangent behavior
Both the function and derivative exist for every real x. Since exp(x²) is positive, the sign of f′ is the sign of x: the function decreases for x<0 and increases for x>0. Its minimum is f(0)=1, with horizontal tangent y=1.
Second derivative
Differentiate once more using the product rule: f″(x)=2exp(x²)+4x²exp(x²)=(2+4x²)exp(x²). This is positive everywhere, so the graph is concave up on the real line.
Common mistake
Do not use the power rule on e^(x²), or replace the exponent by 2x. The derivative of the exponent is a multiplier.
Practice problems
Differentiate exp(3x²)
Answer: 6x·exp(3x²): multiply by the derivative of 3x².
Differentiate exp(x²+1)
Answer: 2x·exp(x²+1): the added constant has derivative zero.
Why e^(x²) is not the same as e^(2x)
The placement of the square changes the function completely. In e^(x²), the entire variable expression x² is the exponent, so its derivative contributes the variable factor 2x. By contrast, (e^x)²=e^(2x), whose inner exponent is 2x and whose derivative is the constant 2. The first derivative is 2x·e^(x²); the second is 2e^(2x). Parentheses are therefore part of the mathematics, not optional typography.
A numerical and graphical check
At x=1, the formula gives f′(1)=2e. A centered secant calculation, [e^((1+h)²)−e^((1−h)²)]/(2h), approaches 2e as h becomes small. At x=−1 the derivative is −2e. These opposite slopes agree with the graph’s symmetry: e^(x²) is even, while its derivative 2x·e^(x²) is odd. The curve falls toward its minimum on the left and rises away from it on the right.
Growth and higher derivatives
The function grows faster than e^x as x tends to positive infinity because x² eventually exceeds x. It also grows without bound as x tends to negative infinity, unlike e^x, because squaring makes the exponent positive. The second derivative (2+4x²)e^(x²) is positive for every real input. That proves strict concavity upward and confirms that x=0 is the unique global minimum.
More worked variations
e^(4x²): let u=4x², so u′=8x and the derivative is 8x·e^(4x²).
e^((x−2)²): the inner derivative is 2(x−2), giving 2(x−2)e^((x−2)²). The horizontal tangent moves from x=0 to x=2.
x·e^(x²): combine product and chain rules: e^(x²)+2x²e^(x²)=(1+2x²)e^(x²).
Questions students ask
Can the 2x move into the exponent?
No. Multiplication outside an exponential is not interchangeable with addition inside its exponent. Keep 2x as an external factor.
Does e^(x²) ever equal zero?
No. A real exponential is always positive. The derivative is zero at x=0 because of the factor 2x, not because the exponential vanishes.
Related exponential derivatives
Review the chain rule and product rule, or read our verification method and limitations.