Derivative of e^(−x)
A negative exponent changes the sign of the slope. This example separates exponential decay from a negative function value.
Step-by-step solution
Let u=−x. Its derivative is −1.
The derivative of exp(u) with respect to u is exp(u).
f′(x)=exp(−x)·(−1)=−exp(−x).
Domain and tangent behavior
The function is positive for every real x, but its derivative is negative everywhere. It is strictly decreasing. At x=0 the point is (0,1), its slope is −1, and its tangent line is y=1−x.
Second derivative
The second derivative is exp(−x), which is positive. Thus a decreasing function can still be concave up. As x tends to positive infinity, exp(−x) tends to zero without reaching it.
Common mistake
A negative exponent does not make the exponential negative. The minus sign in the derivative comes from differentiating −x.
Practice problems
Differentiate exp(−3x)
Answer: −3exp(−3x): the inner derivative is −3.
Differentiate 5exp(−2x)
Answer: −10exp(−2x): retain the constant 5 and multiply by −2.
Related exponential derivatives
Review the chain rule and product rule, or read our verification method and limitations.