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The chain rule is a fundamental technique in calculus used to differentiate composite functions. A composite function is one in which one function is nested inside another. For example, in the function \(y = e^{3x}\), we have the exponential function \(e^u\), where \(u = 3x\). The chain rule helps us differentiate such functions efficiently. To apply the chain rule, we follow these steps:

• Differentiate the outer function with respect to its inner function.
• Multiply by the derivative of the inner function.

Applying this to \(y = e^{3x}\), the derivative of \(e^u\) with respect to \(u\) is \(e^u\), and the derivative of \(3x\) with respect to \(x\) is \(3\). Thus, the derivative using the chain rule is \(3e^{3x}\). This results in the formula \(y'(x) = 3e^{3x}\), which represents the slope of the tangent at any point \(x\) on the curve.

The derivative of a function represents the rate of change or the slope of the function at any given point. In simple terms, it tells us how steep the function is at a particular point. For the function \(y = e^{3x}\), the derivative was computed as \(y'(x) = 3e^{3x}\). This formula helps us understand how the function grows as \(x\) changes.

To find the exact slope of the tangent line at a specific point, we substitute the point's \(x\)-coordinate into the derivative. In this case, substituting \(x = 0\) into \(y'(x)\) gives us \(3e^{0} = 3\). This means at the point \((0, 1)\), the slope of the tangent line to the graph of \(y = e^{3x}\) is \(3\). A similar process is followed for \(y = e^{-3x}\) to find its slope.

Exponential functions are a type of mathematical function where a constant base, such as \(e\), is raised to a variable exponent. These functions are frequently used in real-world situations like compound interest, population growth, and radioactive decay. The function \(y = e^{3x}\) is an example where the constant base \(e\) is raised to the power of \(3x\).

• The steep increase or decrease behaviour of exponential functions is due to the exponent, which leads to rapid growth or decay.
• In our exercise, we see both increasing exponential function \(e^{3x}\) and decreasing exponential function \(e^{-3x}\).

The derivative of an exponential function \(a^{x}\) is obtained by multiplying the original function by the natural logarithm of the base, specifically for \(e\), the derivative of \(e^{x}\) is simply \(e^{x}\). This property makes exponential functions very interesting in calculus.

The point-slope form of a linear equation is a method to write the equation of a line if you know the slope and a point on the line. The formula is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.

• It is particularly useful for writing equations of tangent lines, as we often know a specific point and the slope at that point from the derivative.
• The formula rearranges to make \(y\) the subject, giving a classic line equation \(y = mx + b\), where \(b\) is the y-intercept.

For instance, given that the slope \(m = 3\) and point \((0, 1)\), we can use the point-slope form to write the equation of the tangent line to the graph of \(y = e^{3x}\) as \(y - 1 = 3(x - 0)\). Simplifying this gives the equation \(y = 3x + 1\).