Certainly! Let's break down the question and find the correct answer step-by-step.
1. Differentiability and Definition of the Derivative:
If [tex]\( f \)[/tex] is a differentiable function at [tex]\( a \in \text{Domain } f \)[/tex], it means that the derivative of [tex]\( f \)[/tex] at [tex]\( a \)[/tex], denoted [tex]\( f'(a) \)[/tex], exists.
2. Expression for the Derivative:
The expression given in the problem is:
[tex]\[
\frac{f(a + h) - f(a)}{h}
\][/tex]
This expression is known as the difference quotient. It represents the average rate of change of the function [tex]\( f \)[/tex] over the interval from [tex]\( a \)[/tex] to [tex]\( a + h \)[/tex].
3. Limit Process to Define Derivative:
As [tex]\( h \)[/tex] approaches zero (from either the positive or negative direction), the difference quotient approaches the derivative [tex]\( f'(a) \)[/tex]:
[tex]\[
f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}
\][/tex]
This limit, when it exists, gives the instantaneous rate of change of the function [tex]\( f \)[/tex] at [tex]\( a \)[/tex].
4. Interpreting the Derivative:
The derivative [tex]\( f'(a) \)[/tex] at a point [tex]\( (a, f(a)) \)[/tex] geometrically represents the gradient or slope of the tangent line to the graph of [tex]\( f \)[/tex] at that point.
5. Analyzing Each Choice:
- Choice A: The gradient of the normal line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is not correct since the normal line is perpendicular to the tangent line, and its gradient is related to but not the same as [tex]\( f'(a) \)[/tex].
- Choice B: The gradient of the secant line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is incorrect. The secant line connects two points on the graph, and its gradient is given by the difference quotient without taking the limit.
- Choice C: The gradient of the vertical line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is incorrect. A vertical line has an undefined gradient.
- Choice D: The gradient of the tangent line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex] — This is correct since the derivative [tex]\( f'(a) \)[/tex] gives the gradient of the tangent line at [tex]\( (a, f(a)) \)[/tex].
6. Conclusion:
Therefore, the expression [tex]\(\frac{f(a + h) - f(a)}{h}\)[/tex] as [tex]\( h \)[/tex] approaches 0 from both directions gives the derivative [tex]\( f'(a) \)[/tex], which is the gradient of the tangent line to the graph of [tex]\( f \)[/tex] at the point [tex]\( (a, f(a)) \)[/tex].
The correct answer is:
[tex]\[
\boxed{\text{D. The gradient of the tangent line to the graph of f at a point } ( a , f ( a ))}
\][/tex]
