Basic Differentiation 00D5Cb

1. Let's start by understanding the problem: you want to learn about differentiation, which is a fundamental concept in calculus used to find the rate at which a function changes. 2. The basic formula for differentiation of a function $f(x)$ is given by the limit definition: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This formula calculates the derivative $f'(x)$, which represents the slope of the tangent line to the curve at any point $x$. 3. Important rules to remember: - The derivative of a constant is 0. - The power rule: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. - The sum rule: the derivative of a sum is the sum of the derivatives. 4. Let's differentiate an example function step-by-step: $f(x) = 3x^2 + 5x - 4$. 5. Using the power rule: - Derivative of $3x^2$ is $3 \times 2x^{2-1} = 6x$. - Derivative of $5x$ is $5$. - Derivative of $-4$ is $0$. 6. So, the derivative is: $$f'(x) = 6x + 5$$ 7. This means the rate of change of the function $f(x)$ at any point $x$ is $6x + 5$. 8. Differentiation helps in many fields like physics for velocity and acceleration, economics for cost and revenue analysis, and more. Feel free to ask for more examples or specific rules!