At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
If you're just starting calculus, perhaps you're asking about using the definition of the derivative to differentiate [tex]x^4[/tex].
We have
[tex]\dfrac{d}{dx} x^4 = \displaystyle \lim_{h\to0} \frac{(x+h)^4 - x^4}h[/tex]
Expand the numerator using the binomial theorem, then simplify and compute the limit.
[tex]\dfrac{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}[/tex]
In general, the derivative of a power function [tex]f(x) = x^n[/tex] is [tex]\frac{df}{dx} = nx^{n-1}[/tex]. (This is the aptly-named "power rule" for differentiation.)
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
