How to calculate the derivative of a given function?

To calculate the derivative of a given function, you'll use basic differentiation rules. The derivative of a function at a specific point is the slope of the tangent line to the curve at that point. Here's how to find derivatives using basic rules:

1. Constant Rule: The derivative of a constant is 0.

   • Example: If $f(x) = 5$, then $f'(x) = 0$.

2. Power Rule: For a function $f(x) = x^n$, where $n$ is a constant, the derivative is given by $f'(x) = nx^{n-1}$.

   • Example: If $f(x) = x^4$, then $f'(x) = 4x^3$.

3. Sum and Difference Rules: The derivative of a sum (or difference) of functions is the sum (or difference) of their derivatives.

   • Example: If $f(x) = x^2 + 3x - 2$, then $f'(x) = 2x + 3$.

4. Constant Multiple Rule: The derivative of a constant multiple of a function is the constant times the derivative of the function.

   • Example: If $f(x) = 2x^3$, then $f'(x) = 6x^2$.

5. Product Rule: For functions $u(x)$ and $v(x)$, the derivative of their product is given by:

$$(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)$$

• Example: If $u(x) = x^2$ and $v(x) = 3x$, then $(u(x)v(x))' = (x^2)(3x)' + (x^2)'(3x) = 3x^2 + 6x$.

1. Quotient Rule: For functions $u(x)$ and $v(x)$, the derivative of their quotient is given by:

$$\left(\frac{u(x)}{v(x)}\right)' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$$

• Example: If $u(x) = x^2$ and $v(x) = 3x$, then $\left(\frac{u(x)}{v(x)}\right)' = \frac{(x^2)'(3x) - (x^2)(3x)'}{(3x)^2} = \frac{6x - 3x^2}{9x^2}$.