How to find slope of exponential function?
Understand the Problem
The question is asking how to determine the slope of an exponential function, which typically involves taking the derivative of the function to find its rate of change at a given point.
Answer
The slope of the function at point \( x_0 \) is given by \( f'(x_0) = a \cdot b^{x_0} \cdot \ln(b) \).
Answer for screen readers
The slope of an exponential function ( f(x) = a \cdot b^x ) at a specific point ( x_0 ) is given by:
$$ f'(x_0) = a \cdot b^{x_0} \cdot \ln(b) $$
Steps to Solve
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Identify the exponential function
Start with the general form of an exponential function, which is usually given as ( f(x) = a \cdot b^x ), where ( a ) is a constant, ( b ) is the base of the exponent (and usually ( b > 0 )), and ( x ) is the variable. -
Differentiate the function
To find the slope of the function, we need to take the derivative. The derivative of an exponential function ( f(x) = a \cdot b^x ) is given by:
$$ f'(x) = a \cdot b^x \cdot \ln(b) $$
This formula helps us determine the rate of change of the function. -
Evaluate the derivative at a specific point
If a specific point ( x_0 ) is given, substitute that value into the derivative to find the slope at that point:
$$ \text{slope} = f'(x_0) = a \cdot b^{x_0} \cdot \ln(b) $$ -
Interpret the result
The result from the last step gives you the slope of the exponential function at the specified point. A positive slope indicates the function is increasing at that point, while a negative slope indicates it is decreasing.
The slope of an exponential function ( f(x) = a \cdot b^x ) at a specific point ( x_0 ) is given by:
$$ f'(x_0) = a \cdot b^{x_0} \cdot \ln(b) $$
More Information
The slope of an exponential function increases as ( x ) increases, which means as you move along the curve, the steepness of the slope becomes greater. This is a key characteristic of exponential functions, showing their rapid growth or decay depending on the base ( b ).
Tips
- Misidentifying the form of the exponential function or forgetting to include the constant ( a ).
- Confusing the rules of differentiation, particularly failing to apply the chain rule or correctly using the properties of logarithms.
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