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Well, isn't that a happy little question! To find the derivative of (x-1)^x, we'll need to use logarithmic differentiation. Start by taking the natural logarithm of both sides, then apply implicit differentiation to find the derivative. Remember, there are no mistakes, just happy little accidents in math!
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DevinOh, dude, the derivative of (x-1)^x is a bit of a wild ride. You gotta use logarithmic differentiation for this bad boy. So, first, you take the natural log of both sides, then you apply the chain rule and the product rule like it's a math party. And voilà, you get the derivative! Easy peasy... kinda.
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What is the derivative of the square root of cos x?
To find the derivatve of the square root of cos x: Use the chain rule; this means multiply the inner derivative by the outer derivative. You can write the question f(x) = (cos x)1/2 This general break-down explains how to find d/dx f(x) note: d/dx basically symbolizes "the derivative of" In general terms: f(x) = x1/2 g(x) = cos x f(g(x)) = (cos x)1/2 outer derivative: d/dx f(z) = (1/2)*x-1/2 = 1/(4cos x)1/2 inner derivative: d/dx g(x) = -sin(x) final answer: d/dx f(g(x)) = -sin(x)/(4*cos x)1/2 note: d/dx means "the derivative of"; so d/dx x = 1 Further explained: Set up the equation to a more general form: (cos x)1/2 To make the inner derivative, look at cos(x) To make the outer derivative, look at x1/2 note: x ~ cos x; so we treat (cos x) simply as x, to create the outer derivative You probably know the necessary derivates: 1. derivative of cos x = -sin x 2. derivative of a1/2 = (1/2)*a-1/2 = 1/(4a)1/2 Multiplying the two we get the answer: -sin(x)/(4cos x)1/2
How do you find the rate of change--or slope--in a two variable equation?
For a straight line, if A = (x1 , y1) and B = (x2 , y2) are any two points on the line, then the slope is (y2 - y1)/(x2 - x1) provided x2 is not the same as x1. More generally, if the equation is y = f(x) then the rate of change in y is dy/dx or f'(x), the derivative of the function f(x).
How do you differentiate sin rootx?
By the chain rule, the derivative of sin(x1/2) will be the derivative of x1/2 multiplied by the derivative of the enclosing sine function. Thus, y = sin(x1/2) y' = (1/2)*(x-1/2)*cos(x1/2) For further reading, you might want to consult your calculus book on the chain rule. Here is a site that (kind of) explains the chain rule, though it does have good examples: http://archives.math.utk.edu/visual.calculus/2/chain_rule.4/index.html For step-by-step derivatives of functions, try Calc 101: http://calc101.com/webMathematica/derivatives.jsp
What is the derivative of sinπx?
The derivative with respect to 'x' of sin(pi x) ispi cos(pi x)
What if raised to the power of one?
Any number raised to the power 1 is that same number, x1 = x. For example, 51 = 5.
