is the derivative of x 1?
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Explanation: The derivative with respect to x requires a function of x in its definition. x=1 does not describe a function of x . So it is not possible to give a definition of the derivative with respect to x .
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A function is differentiable if it has a derivative. You can think of a derivative of a function as its slope. The relationship between continuous functions and differentiability is-- all differentiable functions are continuous but not all continuous functions are differentiable.
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The first derivative of a curve tells us whether the slope is positive, negative, or horizontal; and the sign (positive or negative) of the second derivative tells us whether the curve is concave up or down. Extrema will be where the slope changes, and Inflection points will be where the concavity changes.
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1 Answer. This function is neither even nor odd, since neither f(−x)=f(x) nor f(−x)=−f(x) holds.
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Thus, the derivative is a slope. ... This is the same as saying that the derivative is the slope of the tangent line to the graph of the function at the given point. The slope of a secant line (line connecting two points on a graph) approaches the derivative when the interval between the points shrinks down to zero.
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1 Answer. The inverse of f(x)=3x+1 is f(x)−1=13x−13 .
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O(1) and Θ(1) aren't necessarily the same if you are talking about functions over real numbers. ... This function is O(1) because for any n ≥ 1, f(n) ≤ 1.
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This cubic function is indeed a "function" as it passes the vertical line test. In addition, this function possesses the property that each x-value has one unique y-value that is not used by any other x-element. This characteristic is referred to as being a 1-1 function.
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The chain rule states that the derivative of the composite function is the product of the derivative of f and the derivative of g. ... This model may not have a constant derivative.
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Integration goes the other way: the integral (or antiderivative) of 1/x should be a function whose derivative is 1/x. As we just saw, this is ln(x). However, if x is negative then ln(x) is undefined!
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Yes, they are the same. You just have to factor the (x2 - 1) in the numerator of f(x) and cancel out the (x + 1).
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Domain: So the domain of f(x) := tanx is all real numbers except x = π 2 + kπ, k an integer. All of the trig functions are periodic and thus are not one-to-one.
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⎢⎝ x x > 0 0 x = 0 −x x < 0 . ... The left limit does not equal the right limit, and therefore the limit of the difference quotient of f(x) = |x| at x = 0 does not exist. Thus the absolute value function is not differentiable at x = 0.
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Verify that the partial derivative Fxy is correct by calculating its equivalent, Fyx, taking the derivatives in the opposite order (d/dy first, then d/dx). In the above example, the derivative d/dy of the function f(x,y) = 3x^2*y - 2xy is 3x^2 - 2x.
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Example: Square and Square Root (continued) of each other about the diagonal y=x. Note: when we restrict the domain to x ≤ 0 (less than or equal to 0) the inverse is then f-1(x) = −√x: {x2 | x ≤ 0 } {−√x | x ≥ 0 }
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Divide x x by 1 1 . Since g(f(x))=x g ( f ( x ) ) = x , f−1(x)=ln(x)ln(3) f - 1 ( x ) = ln ( x ) ln ( 3 ) is the inverse of f(x)=3x f ( x ) = 3 x .
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Bill K. While this is technically only true for x≠0 , an interesting thing about this example is that its discontinuity and lack of differentiability at x=0 can be "removed". Let f(x)=sin(x)x . ... Moreover, it's also differentiable everywhere and f'(0)=0 .