# What does differentiable mean for a function?

geometrically, the function

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A function is considered differentiable at a point if it has a derivative at that point. Geometrically, this means that the function has a well-defined tangent line at that point, and the derivative represents the slope of that tangent line. Mathematically, a function ( f(x) ) is differentiable at a point ( x = a ) if the following limit exists:

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

If this limit exists, then the function is differentiable at ( x = a ). If a function is differentiable at every point in its domain, it is called differentiable over its entire domain.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the equation of the tangent line to the curve #x^2 + y^2= 169# at the point (5,-12)?
- How do you find the equation of the line tangent to #f(x)= x^2+3x-7# at x =1?
- What is the slope of the line normal to the tangent line of #f(x) = x^2e^(x-1)+2x # at # x= 1 #?
- For #f(x)=1/x# what is the equation of the tangent line at #x=1/2#?
- How do you find the instantaneous rate of change of the function #f(x) = x^2 + 3x + 4# when #x=2#?

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