Question

Using the definition of a derivative, what is y'(x) given that y(x)=ˣ⁴+ˣ²-5x+3?

Answer 1

The derivative of the function y(x) = x⁴ + x² - 5x + 3, found by applying the power rule to each term, is y'(x) = 4x³ + 2x - 5.

Step-by-step explanation:

To find the derivative y'(x) of the function y(x) = x⁴ + x² - 5x + 3 using the definition of a derivative, you would typically use the limit process. However, for polynomial functions like this one, we can directly apply the power rule for each term. Differentiating each term separately, we have:

• The derivative of x⁴ with respect to x is 4x³.
• The derivative of x² with respect to x is 2x.
• The derivative of -5x with respect to x is -5.
• The derivative of the constant 3 with respect to x is 0 because the derivative of a constant is always zero.

Combining these results, the derivative y'(x) is therefore 4x³ + 2x - 5.