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.
Related exponential derivatives
Review the chain rule and product rule, or read our verification method and limitations.