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Differentiate the function f(x) = sin(2x^2 + 3x).
Published on October 20, 2024
Calculus
The derivative of sin(2x² + 3x) is found using the chain rule, differentiating the outer sine function and the inner quadratic function separately, and then multiplying the results: cos(2x² + 3x) * (4x + 3).
Question
Differentiate f(x) = sin(2x^2 + 3x)
Answer
Let’s break down how to differentiate the function f(x) = sin(2x² + 3x) step-by-step.
1. Identify the Outer and Inner Functions
This function is a composition of two functions:
- Outer function: sin(u)
- Inner function: u = 2x² + 3x
2. Apply the Chain Rule
The chain rule states:
d(f(g(x)))/dx = f’(g(x)) * g’(x)
d(f(g(x)))/dx = f’(g(x)) * g’(x)
In our case:
- f(g(x)) = sin(2x² + 3x)
- g(x) = 2x² + 3x
3. Differentiate the Outer Function
The derivative of sin(u) with respect to u is cos(u).
So, f’(u) = cos(u)
4. Differentiate the Inner Function
The derivative of 2x² + 3x with respect to x is:
g’(x) = 4x + 3
5. Substitute and Simplify
Now, substitute the results from steps 3 and 4 back into the chain rule:
f’(x) = cos(u) * (4x + 3)
Finally, replace ‘u’ with the inner function (2x² + 3x):
f’(x) = cos(2x² + 3x) * (4x + 3)
Therefore, the derivative of f(x) = sin(2x² + 3x) is f’(x) = cos(2x² + 3x) * (4x + 3).