Equation of the tangent for the curve \(\mathrm{y}=\mathrm{a} \log \sec (\mathrm{x} / \mathrm{a})\) at \(\mathrm{x}=\mathrm{a}\) is (a) \((\mathrm{y}-\mathrm{a} \log \sec 1) \tan 1=\mathrm{x}-\mathrm{a}\) (b) \((\mathrm{x}-\mathrm{a}) \tan 1=(\mathrm{y}-\mathrm{a} \log \sec 1)\) (c) \((\mathrm{x}-\mathrm{a}) \cos 1=((\mathrm{y}-\mathrm{a}) \log \sec 1) \tan 1\) (d) None of these

Understanding differentiation is essential in calculus and it refers to finding the derivative of a function with respect to a variable. Derivatives represent the rate at which a function is changing at any given point and are foundational to understanding changes within various systems – from physics to finance.

In the context of the equation of a tangent line, differentiation allows us to find the slope of the tangent at a particular point on a curve. The derivative of the function at a specific value of x gives us precisely this slope. Our goal with differentiation in this scenario is to calculate how steeply the curve is rising or falling at the point where the tangent touches the curve.

The chain rule is a fundamental theorem in calculus that allows us to compute the derivative of composite functions. When a function is the result of other functions working within it (like a 'chain' of functions), this rule allows us to differentiate that function with respect to the variable of interest.

For example, if we have a function like \(f(g(x))\), the chain rule tells us how to differentiate it: \(f'(g(x)) \cdot g'(x)\). It's like peeling an onion, where we start by differentiating the outer layer and then move inward, each time multiplying by the derivative of the next inner layer. This rule is particularly useful when dealing with functions inside of other functions, such as trigonometric or logarithmic functions.

The slope of a tangent is a measure of how steeply a tangent line is inclined with respect to the horizontal axis. In graphical terms, it represents the inclination of the line that just 'touches' the curve at a particular point without cutting through it.

Mathematically, the slope of the tangent line at a given point is equivalent to the value of the derivative of the function at that point. A steeper tangent has a higher slope value, while a flatter tangent has a lower slope value. The beauty of differentiation is that it provides us with this exact slope value, allowing us to precisely determine the direction and steepness of the tangent line.

The point-slope form of a line is a straightforward method to write the equation of a line when we know a specific point on the line, and its slope. The point-slope form equation is: \(y - y_1 = m(x - x_1)\), where \( (x_1, y_1) \) is a point on the line, and \(m\) is the slope of the line.

This form is particularly useful for constructing the equation of a tangent line to a curve at a given point. Once we have determined the slope of the tangent, \(m\), using differentiation, and we have a specific point on the curve, \( (x_1, y_1) \) through which the tangent line passes, we can subsitute these values into the point-slope equation to obtain the explicit equation for the tangent line.