Final answer:
The derivatives are found using the power rule, producing the rates of change for each given function, with constants differentiating to zero and the coefficient rule applying to functions multiplied by constants.
Step-by-step explanation:
Finding the derivative of a function involves applying the rules of differentiation to obtain the function's rate of change or slope at any given point. Let's find the derivatives for the functions provided:
- The derivative of y = x¹⁰ is dy/dx = 10x⁹.
- The derivative of a constant function like y = 50 is always 0.
- For y = 4x⁶, the derivative is dy/dx = 24x⁵.
- With respect to v, the derivative of u = 3v¹² is du/dv = 36v¹¹.
- The derivative of a linear function like y = 7x - 3 is the constant 7.
- Similarly, the derivative of -2x - 4 is -2.
- The derivative of 9x³ - 2 is 27x².
- For x² - 1, the derivative is 2x.
- To find the derivative of the product (9x³ - 2)(x + 1), we can first expand the product and differentiate, or use the product rule.
- Last one, x²², the derivative is dy/dx = 22x²¹.
These examples apply the power rule, one of the basic rules of differentiation. Note that the derivative of a constant is zero, and the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. In the case of u = 3v¹², it's important to respect the variables and differentiate with respect to v.