Answer
See the explanation
Work Step by Step
Certainly! When calculating derivatives, there are several rules that can be applied to find the derivative of a function. Here are some of the key rules:
1. Power Rule: If \(f(x) = x^n\), then \(f'(x) = nx^{(n-1)}\).
2. Constant Rule: If \(f(x) = c\), where \(c\) is a constant, then \(f'(x) = 0\).
3. Sum/Difference Rule: If \(f(x) = g(x) + h(x)\), then \(f'(x) = g'(x) + h'(x)\). Similarly, if \(f(x) = g(x) - h(x)\), then \(f'(x) = g'(x) - h'(x)\).
4. Product Rule: If \(f(x) = g(x) \cdot h(x)\), then \(f'(x) = g'(x) \cdot h(x) + g(x) \cdot h'(x)\).
5. Quotient Rule: If \(f(x) = \frac{g(x)}{h(x)}\), then \(f'(x) = \frac{g'(x) \cdot h(x) - g(x) \cdot h'(x)}{(h(x))^2}\).
6. Chain Rule: If \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x)) \cdot h'(x)\).
Examples:
1. \(f(x) = 3x^2\)
Applying the Power Rule: \(f'(x) = 2 \cdot 3x^{(2-1)} = 6x\).
2. \(g(x) = 5\)
Applying the Constant Rule: \(g'(x) = 0\).
3. \(h(x) = 2x^3 - 4x\)
\(h'(x) = 3 \cdot 2x^{(3-1)} - 4 = 6x^2 - 4\).
4. \(j(x) = x^2 \cdot \sin(x)\)
Applying the Product Rule: \(j'(x) = 2x \cdot \sin(x) + x^2 \cdot \cos(x)\).
Remember to always check for the validity of these rules based on the domain of the function.
