Problem 21 - Single Variable Calculus

Question

find the derivative of the function. \(g(x)=\tanh (1-3 x)\)

Step-by-Step Solution

Answer

The derivative of the function \(g(x) = \tanh (1-3 x)\) is: $$g'(x) = -\frac{12}{(e^{1 - 3x} + e^{3x - 1})^2}$$

Step 1: Apply the chain rule to differentiate \(g(x)\)

The chain rule states that if we have a composite function, \(g(x) = f(h(x))\), then the derivative of \(g(x)\) with respect to \(x\) is given by \(g'(x) = f'(h(x))\cdot h'(x)\). In our case, we have \(g(x) = \tanh(1 - 3x)\). Let \(h(x) = 1 - 3x\) and apply the chain rule to differentiate \(g(x)\).

Step 2: Differentiate the hyperbolic tangent function

Recall that the hyperbolic tangent function is defined as \(\tanh(x) = \dfrac{e^x - e^{-x}}{e^x + e^{-x}}\). Using the quotient rule, we can differentiate \(\tanh(x)\) with respect to \(x\): $$\frac{d}{dx}[\tanh(x)] = \frac{(e^x + e^{-x})(e^x + e^{-x}) - (e^x - e^{-x})(e^x - e^{-x})}{(e^x + e^{-x})^2}$$ Simplify the numerator: $$\frac{(e^{2x} + 2 + e^{-2x}) - (e^{2x} - 2 + e^{-2x})}{(e^x + e^{-x})^2} = \frac{4}{(e^x + e^{-x})^2}$$ So, the derivative of the hyperbolic tangent function is: $$\frac{d}{dx}[\tanh(x)] = \frac{4}{(e^x + e^{-x})^2}$$

Step 3: Compute the derivative of the inner function \(h(x)\)

In our case, the inner function is \(h(x) = 1 - 3x\). To compute its derivative, we have: $$\frac{d}{dx}[1 - 3x] = -3$$

Step 4: Apply the chain rule

Now, we can apply the chain rule to find the derivative of \(g(x)\): $$g'(x) = \frac{d}{dx}[\tanh(1 - 3x)] = \frac{4}{(e^{1 - 3x} + e^{-(1 - 3x)})^2} \cdot (-3) $$ Simplify the expression: $$g'(x) = -\frac{12}{(e^{1 - 3x} + e^{3x - 1})^2}$$ So, the derivative of \(g(x) = \tanh (1-3 x)\) is: $$g'(x) = -\frac{12}{(e^{1 - 3x} + e^{3x - 1})^2}$$

Key Concepts

Chain Rule in Differentiation

The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function is like a "function within a function." For example, if a function is expressed as \(g(x) = f(h(x))\), it means \(f(x)\) operates on \(h(x)\). To find the derivative of \(g(x)\), the chain rule provides a straightforward method.

• First, identify the outer function \(f\) and the inner function \(h\).
• Differentiate the outer function with respect to the inner function as if the inner part were just a simple variable. This gives \(f'(h(x))\).
• Then, differentiate the inner function \(h(x)\) with respect to \(x\), yielding \(h'(x)\).
• Finally, multiply these derivatives, resulting in \(g'(x) = f'(h(x)) \cdot h'(x)\).

In our exercise, the function \(g(x) = \tanh(1 - 3x)\) requires us to apply the chain rule because it consists of the hyperbolic tangent \(\tanh()\) acting on another function \(1 - 3x\). Identifying these pieces is the first step to successfully find the derivative.

Understanding Hyperbolic Tangent

The hyperbolic tangent function, known as \(\tanh(x)\), plays a crucial role in certain areas of mathematics and engineering. It resembles the tangent function from trigonometry, but it is derived from hyperbolic identities. The formula for \(\tanh(x)\) is given by:\[\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]This formula incorporates the exponential function, making it unique from regular trigonometric functions. When differentiating \(\tanh(x)\), the quotient rule is necessary due to its fraction form.The derivative can be determined by applying the quotient rule, resulting in:\[\frac{d}{dx}[\tanh(x)] = \frac{4}{(e^x + e^{-x})^2}\]This result shows the symmetry and properties unique to hyperbolic functions. It's important to appreciate how the derivative of \(\tanh(x)\) differs from that of the ordinary tangent, offering smoother and different growth patterns.

Quotient Rule Simplified

The quotient rule is used for differentiating functions presented as a ratio, \(\frac{u(x)}{v(x)}\). This rule comes in handy when dealing with functions like \(\tanh(x)\). The key steps to apply the quotient rule are:

• Identify \(u(x)\) and \(v(x)\).
• Calculate the derivatives \(u'(x)\) and \(v'(x)\).
• Use the quotient formula: \(\frac{d}{dx}\left[\frac{u(x)}{v(x)}\right] = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}\).

In the case of \(\tanh(x)\), where \(u(x) = e^x - e^{-x}\) and \(v(x) = e^x + e^{-x}\), the application of the quotient rule results in a neat expression for the derivative. Ensuring that each part is clearly identified and calculated smoothly can simplify what initially seems complex.

Basics of Differentiation

Differentiation is a core concept of calculus, focusing on how to find the rate at which a function is changing at any given point. It's the process of determining the derivative, which tells us the slope of the function's graph at any point along its curve.Different functions might require specific rules for differentiation:

• For simple power functions \(f(x) = x^n\), the power rule \(f'(x) = nx^{n-1}\) is used.
• Product rules and quotient rules apply to products and quotients of functions respectively, like \(\tanh(x)\) that uses both compound differentiation and quotient operations.
• The chain rule is essential when dealing with composite functions, as applied in the present exercise with \(\tanh(1-3x)\).

Understanding each approach helps tackle complex calculus problems by breaking them into manageable steps and ensuring a clear path from the original function to its derivative.