 ##### Asked by: Zulmira Agin
science space and astronomy

# How do you find the derivative of a sigmoid function?

Last Updated: 29th March, 2020

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The derivative of the sigmoid is ddxσ(x)=σ(x)(1−σ(x)). Consider f(x)=1σ(x)=1+e−x.

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Herein, what is a derivative of a function?

The derivative measures the steepness of the graph of a function at some particular point on the graph. Thus, the derivative is a slope. The slope of a secant line (line connecting two points on a graph) approaches the derivative when the interval between the points shrinks down to zero.

Subsequently, question is, what is the use of sigmoid function? The main reason why we use sigmoid function is because it exists between (0 to 1). Therefore, it is especially used for models where we have to predict the probability as an output. Since probability of anything exists only between the range of 0 and 1, sigmoid is the right choice. The function is differentiable.

Herein, how do you find the derivative of a logistic function?

Differentiation of logistic function. The logistic function is g(x)=11+e−x, and it's derivative is g′(x)=(1−g(x))g(x). Now if the argument of my logistic function is say x+2x2+ab, with a,b being constants, and I derive with respect to x: (11+e−x+2x2+ab)′, is the derivative still (1−g(x))g(x)?

What is sigmoid derivative?

The derivative of the sigmoid is ddxσ(x)=σ(x)(1−σ(x)). Here's a detailed derivation: ddxσ(x)=ddx[11+e−x]=ddx(1+e−x)−1=−(1+e−x)−2(−e−x)=e−x(1+e−x)2=11+e−x ⋅e−x1+e−x=11+e−x ⋅(1+e−x)−11+e−x=11+e−x ⋅(1+e−x1+e−x−11+e−x)=11+e−x ⋅(1−11+e−x)=σ(x)⋅(1−σ(x))

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