Differentiate.
[tex]f(x)=e^{x^2+3 x}[/tex]
[tex]f^{\prime}(x)=[/tex]
Answer (1)
Identify the function as a composite function and prepare to apply the chain rule.
Differentiate the outer function e u with respect to u , resulting in e u , and substitute u = x 2 + 3 x .
Differentiate the inner function x 2 + 3 x with respect to x , obtaining 2 x + 3 .
Multiply the results to find the derivative: ( 2 x + 3 ) e x 2 + 3 x .
Explanation
Problem Analysis We are given the function f ( x ) = e x 2 + 3 x and asked to find its derivative f ′ ( x ) . This requires us to use the chain rule.
Applying the Chain Rule The chain rule states that if we have a composite function f ( g ( x )) , then its derivative is f ′ ( g ( x )) ⋅ g ′ ( x ) . In our case, we can consider f ( u ) = e u and g ( x ) = x 2 + 3 x . Then f ( g ( x )) = e x 2 + 3 x .
Differentiating the Outer Function First, we find the derivative of the outer function f ( u ) = e u , which is simply f ′ ( u ) = e u . So, f ′ ( g ( x )) = e x 2 + 3 x .
Differentiating the Inner Function Next, we find the derivative of the inner function g ( x ) = x 2 + 3 x . Using the power rule, we have g ′ ( x ) = 2 x + 3 .
Combining the Results Now, we multiply the derivatives together: f ′ ( x ) = f ′ ( g ( x )) ⋅ g ′ ( x ) = e x 2 + 3 x ⋅ ( 2 x + 3 ) . Therefore, f ′ ( x ) = ( 2 x + 3 ) e x 2 + 3 x .
Final Answer Thus, the derivative of the function f ( x ) = e x 2 + 3 x is f ′ ( x ) = ( 2 x + 3 ) e x 2 + 3 x .
Examples
In physics, this type of derivative can be used to describe the rate of change of a quantity that grows exponentially, such as the charge on a capacitor in an RC circuit or the population growth in a biological system. Understanding how to differentiate exponential functions is crucial for modeling and analyzing these phenomena.
