Find the Derivative - d/dx y = natural log of (1+e^x)/(1-e^x)

Problem

d()/d(x)*ln((1+ex)/(1−ex))

Solution

1. Apply logarithm properties to simplify the expression before differentiating. Use the rule ln(a/b)=ln(a)−ln(b)

y=ln(1+ex)−ln(1−ex)

2. Differentiate the first term using the chain rule d(ln(u))/d(x)=1/u⋅d(u)/d(x)

d(ln(1+ex))/d(x)=1/(1+ex)⋅ex

3. Differentiate the second term using the same chain rule.

d(ln(1−ex))/d(x)=1/(1−ex)⋅(−ex)

4. Combine the derivatives and simplify the signs.

d(y)/d(x)=(ex)/(1+ex)−(−ex)/(1−ex)

d(y)/d(x)=(ex)/(1+ex)+(ex)/(1−ex)

5. Find a common denominator to combine the fractions. The common denominator is (1+ex)*(1−ex)=1−e(2*x)

d(y)/d(x)=(ex*(1−ex)+ex*(1+ex))/(1−e(2*x))

6. Distribute and simplify the numerator.

d(y)/d(x)=(ex−e(2*x)+ex+e(2*x))/(1−e(2*x))

d(y)/d(x)=(2*ex)/(1−e(2*x))

Final Answer

d()/d(x)*ln((1+ex)/(1−ex))=(2*ex)/(1−e(2*x))

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