91Ó°ÊÓ

When we talk about derivatives, we're dealing with how a function changes at any given point. The formal definition of a derivative provides a way to find the slope of a tangent line to a function at a particular point. Let's break this down simply. Given a function \( f(x) \), the derivative at a certain point \( x = a \) is defined as:

• \[ f'(a) = \lim_{{h \to 0}} \frac{{f(a + h) - f(a)}}{h} \]

The formula involves a limit process where \( h \) approaches zero. Essentially, you're seeing how the function \( f(x) \) behaves as you nudge \( x \) ever so slightly by \( h \).

In our exercise, we apply this definition to the function \( y = 1 - x^3 \) to find its derivative at \( x = 2 \). By substituting into the formula and simplifying, we conclude that the derivative, which is the slope of the tangent line, is \(-12\). This means, at \( x = 2 \), the function is decreasing quite steeply.

Tangent and normal lines give us a deeper understanding of a curve at a point. The tangent line touches the curve at a point without crossing it, much like a gentle whisper of a touch.

In mathematics, the slope of this tangent line to the curve \( y = 1 - x^3 \) at the point \((2, -7)\) is precisely the value of the derivative at that point. From our derivation, this slope is \(-12\).

• The equation for this tangent line is built using the point-slope form: \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the point on the curve, and \( m \) is the slope.
• For our specific calculation, we have: \[ y + 7 = -12(x - 2) \]
• The complete equation simplifies to: \[ y = -12x + 17 \]

The normal line, however, is perpendicular to the tangent. Its slope is the negative reciprocal of the tangent slope. Hence, for our curve, the normal line's slope is \( \frac{1}{12} \). Accordingly, using the same point, the normal line equation becomes:

• \[ y + 7 = \frac{1}{12}(x - 2) \]
• Or, simplified: \[ y = \frac{1}{12}x - \frac{43}{6} \]