We will use the quotient rule here. express sqrt(2) as 2 1/2.
d/dx[f(x)/g(x) = [g(X)f'(X) - f(x)g'(x)]/(g(x)2
d/dx[t7 * (2 1/2)' - 2 1/2 * (t7)']/(t7)2
= t7 * 0 - 2 1/2 * 7t6/t14
return to sqrt(2)
= 7sqrt(2)t6/t14
============ cleaned up
7sqrt(2)/t8
========
sqrt(x) = x^(1/2) The derivative is (1 / 2) * x^(-1 / 2) = 1 / (2 * x^(1 / 2)) = 1 / (2 * sqrt(x))
-1
y=(8x).5 + (4x).5 = (2+2sqrt(2))x.5 y'=(1 + sqrt(2))/sqrt(x)
4/x can be written as 4x-1 (the power of negative 1 means it is the denominator of the fraction) 4*-1 = -4 Therefore, the derivative is -4x-2
e^(-2x) * -2 The derivative of e^F(x) is e^F(x) times the derivative of F(x)
The derivative of sqrt(2) is zero.
To find the derivative of ( \frac{3}{\sqrt{x}} ), we can rewrite it as ( 3x^{-\frac{1}{2}} ). Using the power rule, the derivative is ( -\frac{3}{2} x^{-\frac{3}{2}} ). This can also be expressed as ( -\frac{3}{2\sqrt{x^3}} ).
Derivative of x = 1, and since sqrt(x) = x^(1/2), derivative of x^(1/2) = (1/2)*(x^(-1/2))Add these two terms together and derivative = 1 + 1/(2*sqrt(x))
sqrt(x) = x^(1/2) The derivative is (1 / 2) * x^(-1 / 2) = 1 / (2 * x^(1 / 2)) = 1 / (2 * sqrt(x))
3x - 4 sqrt(2)The first derivative with respect to 'x' is 3.
The anti-derivative of sqrt(x) : sqrt(x)=x^(1/2) The anti-derivative is x^(1/2+1) /(1/2+1) = (2/3) x^(3/2) The anti-derivative is 4e^x is 4 e^x ( I hope you meant e to the power x) The anti-derivative of -sin(x) is cos(x) Adding, the anti-derivative is (2/3) x^(3/2) + 4 e^x + cos(x) + C
The formula for the derivative of an inverse (finv)' = 1/(f' o (finv)) allows you get a formula for the derivative of the inverse of any function that you already know the derivative of. For example: What is the derivative of sqrt(x)? You could figure this out using the definition of the derivative, but it is complicated. You already know that the derivative of x2 is 2x. So let f = x2; finv = sqrt(x), f' = 2x. This gives: (sqrt(x))' = 1/(2 sqrt(x)). Now you have derived a "square root rule" with almost no work.
sqrt(X) is also X^1/2 use power rule 1/2X^-1/2 ( first derivative ) -1/4X^-3/2 ( second derivative ) and so on
Derivative with respect to 'x' of (5x)1/2 = (1/2) (5x)-1/2 (5) = 2.5/sqrt(5x)
Take the derivative of (3x+1)1/2First use the power rule on the entire function, then multiply it by the derivative of the inside:(1/2)(3x-1)-1/2(3) = (3/2)(3x-1)-1/2which can also be written as3/(2sqrt(3x-1))(sqrt stands for square root)
2 to the power of 2 over 3, or (2^{\frac{2}{3}}), can be expressed as the cube root of (2^2). This simplifies to (\sqrt[3]{4}), which is approximately (1.5874).
By using a form of 1, sqrt(2)/sqrt(2),and multiplying. 15/sqrt(2) * sqrt(2)/sqrt(2) = 15sqrt(2)/2 -------------------