Question

Greetings. As a beginner, I'm struggling a bit to learn calculus. May I know what is the derivative of x to the power 4 step by step?

Thanks in advance!

Answer 1

If you're just starting calculus, perhaps you're asking about using the definition of the derivative to differentiate
`x^4`.

We have

`(d)/(dx) x^4 = \displaystyle \lim_(h\to0) \frac{(x+h)^4 - x^4}h`

Expand the numerator using the binomial theorem, then simplify and compute the limit.

`(d)/(dx) x^4 = \displaystyle \lim_(h\to0) \frac{(x^4+4hx^3 + 6h^2x^2 + 4h^3x + h^4) - x^4}h \\\\ ~~~~~~~~ = \lim_(h\to0) \frac{4hx^3 + 6h^2x^2 + 4h^3x + h^4}h \\\\ ~~~~~~~~ = \lim_(h\to0) (4x^3 + 6hx^2 + 4h^2x + h^3) = \boxed{4x^3}`

In general, the derivative of a power function
`f(x) = x^n` is
`(df)/(dx) = nx^(n-1)`. (This is the aptly-named "power rule" for differentiation.)