# How do you determine the rate of change of a function?

Instantaneous rate of change is the first derivative

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To determine the rate of change of a function, you calculate the derivative of the function with respect to the variable of interest. The derivative represents the rate at which the function's value changes with respect to the independent variable. In calculus, the derivative measures the slope of the function's graph at any given point, indicating how steeply the function is rising or falling. Mathematically, you can find the derivative of a function using differentiation techniques such as the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the function. Once you have the derivative, you evaluate it at a specific point to determine the instantaneous rate of change at that point, or you analyze its behavior across an interval to understand the overall rate of change.

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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.

- Let #f(x) = -3 x^3 + 9 x + 4#, how do you use the limit definition of the derivative to calculate the derivative of f?
- How do you find the instantaneous rate of change of #g(t)=3t^2+6# at t=4?
- How do you find the instantaneous rate of change of the function #x^2 + 3x - 4# when x=2?
- How do you find the equation of the line tangent to #y=sqrt(t-1)# at (5,2)?
- What is the equation of the tangent line of #f(x)=x^2 + sinx # at #x=pi#?

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