Differentiate f(x)=(5x^2-4x)e^x to f’(x)

To differentiate the function \( f(x) = (5x^2 - 4x)e^x \), we need to use the product rule. The product rule states that if you have a function \( f(x) = u(x)v(x) \), then the derivative \( f'(x) \) is given by:

\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]

Here, let \( u(x) = 5x^2 - 4x \) and \( v(x) = e^x \).

First, we find the derivatives of \( u(x) \) and \( v(x) \):

1. \( u(x) = 5x^2 - 4x \)
\[ u'(x) = \frac{d}{dx}(5x^2 - 4x) = 10x - 4 \]

2. \( v(x) = e^x \)
\[ v'(x) = \frac{d}{dx}e^x = e^x \]

Now, apply the product rule:

\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]

Substitute \( u'(x) \), \( u(x) \), \( v(x) \), and \( v'(x) \):

\[ f'(x) = (10x - 4)e^x + (5x^2 - 4x)e^x \]

Factor out \( e^x \):

\[ f'(x) = e^x \left( (10x - 4) + (5x^2 - 4x) \right) \]

Combine the terms inside the parentheses:

\[ f'(x) = e^x (5x^2 + 6x - 4) \]

Thus, the derivative of \( f(x) = (5x^2 - 4x)e^x \) is:

\[ f'(x) = e^x (5x^2 + 6x - 4) \]

Do you want details or have any questions?

Here are 8 related questions to expand your understanding:
1. How do you find the derivative of a product of three functions?
2. What is the chain rule in differentiation and how is it applied?
3. How do you differentiate a composite function?
4. Can you explain the second derivative and its significance?
5. How do you find the derivative of implicit functions?
6. What is the derivative of trigonometric functions?
7. How do you find the derivative of inverse functions?
8. What are higher-order derivatives and how are they useful?

**Tip:** Always double-check your work by simplifying the expression and ensuring all terms are correctly differentiated and combined.

Learn how to differentiate the function f(x) = (5x^2 - 4x)e^x with a step-by-step solution. This problem covers differentiation using the product rule and exponential functions. Suitable for undergraduate students studying calculus.

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