22–36. Derivatives Find the derivatives of the following functions.

f(x) = tanh²x

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Step 1: Recognize that the function f(x) = tanh²(x) is a composite function. It involves the square of the hyperbolic tangent function, tanh(x). To differentiate it, we will use the chain rule.

Step 2: Apply the chain rule. The derivative of tanh²(x) is 2 * tanh(x) * (d/dx of tanh(x)). This is because the outer function is x², and the inner function is tanh(x).

Step 3: Recall the derivative of tanh(x). The derivative of tanh(x) is sech²(x), where sech(x) is the hyperbolic secant function.

Step 4: Substitute the derivative of tanh(x) into the expression from Step 2. This gives: f'(x) = 2 * tanh(x) * sech²(x).

Step 5: Simplify the expression if needed. The derivative of f(x) = tanh²(x) is now fully expressed as f'(x) = 2 * tanh(x) * sech²(x).

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Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when differentiating functions like f(x) = tanh²x, where tanh is a function itself.

Hyperbolic Functions

Hyperbolic functions, such as tanh, sinh, and cosh, are analogs of the trigonometric functions but are based on hyperbolas instead of circles. The function tanh(x) is defined as the ratio of the hyperbolic sine and cosine, and it has unique properties, including its range and behavior as x approaches infinity. Understanding these functions is essential for differentiating expressions involving them.

22–36. Derivatives Find the derivatives of the following functions.f(x) = tanh²x