does anyone know the shortcut to finding the derivative of a function
for example: x^3-3x^2
short cut to what? By inspection, the derivative is 3x^2-6x using the power rule.
Now if you are finding derivatives by the limit (lim (f(x+h)-f(x))/h, you need to work it out.
Yes, there is a shortcut for finding the derivative of a function using the power rule of differentiation. The power rule states that to find the derivative of a function in the form of f(x) = x^n, you multiply the coefficient of x by the exponent n, then decrease the exponent by 1.
To find the derivative of the function f(x) = x^3 - 3x^2, we can apply the power rule separately to each term.
1. Start by differentiating the first term, x^3:
- Multiply the coefficient of x (which is 1) by the exponent (which is 3): 1 * 3 = 3
- Decrease the exponent by 1: 3 - 1 = 2
- So, the derivative of x^3 is 3x^2.
2. Then, differentiate the second term, -3x^2:
- Multiply the coefficient of x^2 (which is -3) by the exponent (which is 2): -3 * 2 = -6
- Decrease the exponent by 1: 2 - 1 = 1
- So, the derivative of -3x^2 is -6x.
To find the derivative of the entire function f(x), we sum up the derivatives of each term:
f'(x) = (3x^2) + (-6x)
Therefore, the derivative of f(x) = x^3 - 3x^2 is f'(x) = 3x^2 - 6x.