I'm not sure if you're asking what the derivative is for f(x) = 1/[(x^2)+2] or for f(x) = [1/(x^2)]+2 so I'm gonna do the first one for now. Lemme know if you want the other one too! :)
**Note: I'm not sure which terminology you're most familiar with using in class but just know that y is the same thing as/another way or writing y(x) or f(x) on the left side of the equation. Similarly, y' = y'(x) = f'(x) = d/dx(y) = dy/dx = d/dx[y(x)] = d/dx[f(x)].**
So we know the First Principles from the Fundamental Theorem of Calculus defines the derivative as a limit:
f'(x) = lim(h→0)⡠[f(x+h)-f(x)]/h
Our function is:
f(x) = 1/[(x^2)+2]
Similarly:
f(x+h) = 1/[((x+h)^2)+2]
Simplify:
f(x+h) = 1/[(x+h)(x+h)+2]
f(x+h) = 1/[(x^2+2xh+h^2+2]
Then we plug it into the limit and simplify:
f'(x) = lim(h→0)⡠〖[1/(((x+h)^2)+2)] - [1/((x^2)+2))]〗/h
f'(x) = lim(h→0)⡠〖[1/(((x+h)^2)+2)] - [1/((x^2)+2))]〗*(1/h)
f'(x) = lim(h→0)⡠〖[1/(((x+h)^2)+2)]*[(x^2)+2]/[(x^2)+2]
- [1/((x^2)+2))]*[((x+h)^2)+2]/[((x+h)^2)+2]〗*(1/h)
f'(x) = lim(h→0)⡠〖[[(x^2)+2]-[((x+h)^2)+2]]/[[(x^2)+2]*[((x+h)^2)+2]]〗*(1/h)
f'(x) = lim(h→0)⡠〖[[(x^2)+2]-[(x^2+2xh+h^2)+2]]/[[(x^2)+2]*[((x^2+2xh+h^2)+2]]〗*(1/h)
f'(x) = lim(h→0)⡠〖[(x^2)+2-(x^2)-2xh-(h^2)-2]/[[(x^2)+2]*[((x^2+2xh+h^2)+2]]〗*(1/h)
f'(x) = lim(h→0)⡠〖[(x^2)+2-(x^2)-2xh-(h^2)-2]/[(x^4)+2(x^3)h+(x^2)(h^2)+2(x^2)+2(x^2)+4xh+2(h^2)+4]〗*(1/h)
f'(x) = lim(h→0)⡠〖[-2xh-(h^2)]/[(x^4)+2(x^3)h+(x^2)(h^2)+4(x^2)+4xh+2(h^2)+4]〗*(1/h)
f'(x) = lim(h→0)⡠(2h)〖(-x-h)/[(x^4)+2(x^3)h+(x^2)(h^2)+4(x^2)+4xh+2(h^2)+4]〗*(1/h)
f'(x) = lim(h→0)⡠(2)〖(-x-h)/[(x^4)+2(x^3)h+(x^2)(h^2)+4(x^2)+4xh+2(h^2)+4]〗
Finally, plug in 0 for h:
f'(x) = lim(h→0)⡠(2)〖(-x-(0))/[(x^4)+2(x^3)(0)+(x^2)((0)^2)+4(x^2)+4x(0)+2((0)^2)+4]〗 f'(x) = (2)[-x/[(x^4)+4(x^2)+4]
Factor the denominator:
f'(x) = (2)[-x/[(x^2)+2)^2]
Final answer: f'(x) = -2x/[((x^2)+2)^2]
Hope this helped!! ~Casey
When the first derivative of the function is equal to zero and the second derivative is positive.
1/e
I'll get you started. Using the definition of the derivative:For f(x) = xsinx this gives:Recall thatFrom here you should be able to finish it out. Post back if you're still having difficulties.
f'(x)=-sin2x(2) f'(x)=-2sin2x First do the derivative of cos u, which is -sin u. Then because of the chain rule, you have to take the derivative of what's inside and the derivative of 2x is 2.
Derivatives are usually taken with respect to time. The first derivative would have units of volume / time, i.e., a flow - for example - "so-and-so many cubic meters per second flow down our river".The second derivate would refer to a change in the flow - when the flow of a liquid or gas increases or decreases with time.
The derivative of sin(x) is cos(x).
First derivative of displacement with respect to time = velocity. Second derivative of displacement with respect to time = acceleration. Third derivative of displacement with respect to time = jerk.
3x - 4 sqrt(2)The first derivative with respect to 'x' is 3.
First derivative of distance with respect to time.
in case of derivative w.r.t time first derivative with a variable x gives velocity second derivative gives acceleration thid derivative gives jerk
Definition: Acceleration is the rate of change of velocity as a function of time. It is vector. In calculus terms, acceleration is the second derivative of position with respect to time or, alternately, the first derivative of the velocity with respect to time.
Position is a vector. Therefore, its first derivative with respect to time (velocity), and its second derivative with respect to time (acceleration) are also vectors.
I'm not sure about the respect to time, but the equation for velocity is the first derivative of the equation of time (w/ respect to distance) and acceleration is the second derivative. I'm sorry, I don't think I properly answered your question, but this information should be correct. . :)
The speed of an object is its rate of displacement. Specifically, it is the rate of displacement over time or the first derivative of displacement with respect to time.
All it means to take the second derivative is to take the derivative of a function twice. For example, say you start with the function y=x2+2x The first derivative would be 2x+2 But when you take the derivative the first derivative you get the second derivative which would be 2
Speed is scalar (it doesn't have direction), and the magnitude of velocity (a vector). The first derivative of velocity is acceleration, therefore the first derivative of speed is the magnitude of acceleration.
2x is the first derivative of x2.