(x23)'=23x22 Generally, (xn)'=n*xn-1
The derivative of sin(x) is cos(x).
if f(x)=kx, f'(x)=ln(k)*kx. Therefore, the derivative of 2x is ln(2)*2x.
Afetr you take the first derivative you take it again Example y = x^2 dy/dx = 2x ( first derivative) d2y/dx2 = 2 ( second derivative)
The derivative of x divided by 3 is 1/3. This can be found using the power rule of differentiation, where the derivative of x^n is nx^(n-1). In this case, x can be written as x^1, so the derivative is 1(1/3)*x^(1-1) = 1/3.
To differentiate y=sin(sin(x)) you need to use the chain rule. A common way to remember the chain rule is "derivative of the outside, keep the inside, derivative of the inside". First, you take the derivative of the outside. The derivative of sin is cos. Then, you keep the inside, so you keep sin(x). Then, you multiple by the derivative of the inside. Again, the derivative of sinx is cosx. In the end, you get y'=cos(sin(x))cos(x))
The first derivative of e to the x power is e to the power of x.
2x is the first derivative of x2.
The derivative is 2x based on the power rule. Multiply the power by the coefficient of x then drop the power by one.
Oh, dude, the third derivative of ln(x) is -2/(x^3). But like, who really needs to know that, right? I mean, unless you're planning on impressing your calculus teacher or something. Just remember, math is like a puzzle, except no one actually wants to put it together.
d/dx(X^4) = 4X^3 ( first derivative ) d/dx(4X^3) = 12X^2 ( second derivative )
The derivative of ex is ex
x^0 = 1 for all x. The derivative of 1 is always zero.
Write sec x as a function of sines and cosines (in this case, sec x = 1 / cos x). Then use the division formula to take the first derivative. Take the derivative of the first derivative to get the second derivative. Reminder: the derivative of sin x is cos x; the derivative of cos x is - sin x.
The fourth derivative of ( \ln(x) ) can be determined by first calculating its derivatives. The first derivative is ( \frac{1}{x} ), the second derivative is ( -\frac{1}{x^2} ), the third derivative is ( \frac{2}{x^3} ), and the fourth derivative is ( -\frac{6}{x^4} ). Thus, the fourth derivative of ( \ln(x) ) is ( -\frac{6}{x^4} ).
e^(-2x) * -2 The derivative of e^F(x) is e^F(x) times the derivative of F(x)
2x
Express the cosecant in terms of sines and cosines; in this case, csc x = 1 / sin x. This can also be written as (sin x)-1. Remember that the derivative of sin x is cos x, and use either the formula for the derivative of a quotient (using the first expression), or the formula for the derivative of a power (using the second expression).