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Rational numbers as exponents Rational numbers as exponents In today’s session we are going to study about derivatives. Derivatives are used in calculus which is a branch of mathematics. A derivative tells us how a function changes. When the input of a given function changes then it makes some changes in function too. To measure these changes we use Derivatives. Simply we can say that derivative measures how much a quantity changes with respect to other quantity. For instance: Velocity; which is a derivative of the position of a moving thing or object with respect to time. The derivative of a function at a very specific or particular point or input value tells us the linear approximation of the function which lies near to that input value. For example we have a real valued function with a single real variable; we have the specific point for derivative is the slope of the tangent line to the graph of the given function. Also the derivative of a given function at a specific point is a linear transformation that is known as the Linearization of the function. To find a derivative of a function we need to do is the differentiation; that is a process reverse process of integration

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So we can say by the fundamental theorem of calculus that the integration and antidifferentiation of a function is same process. As we know that to find the derivative of a function we use a process that is Differentiation; now we will discuss about it. This is basically the rate of change of an output Y which changes with respect to the input X. The output y depends on the input X. The dependency of output upon the input means that the output y is the function of input X. Y = f(X) In the above equation f denotes the function. If the X and Y are the real numbers then the derivative of the f(X), measures the slope of the function at each point. If Y is a linear function of X i.e. Y = M*X + C; then the slope M = change in output Y/ change in input X This formula is true because

M = dy/dx

Y + dY = f(X + dX) = M(X + dX) + C = MX + C + M*dX = Y + MdX. And it tracks the dY = M*dX

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differentiation is used to get the exact value of the rate of change of a function with respect to inputs. In derivatives we use a very important term that is known as inflection point. The inflection point of a function is a point where the second derivative of a function changes its sign. Consider a case where at point the second derivative of a function can be zero if the X = 0 of the function Y = X^3. Similarly in case where function Y = X^1/3, this point may fail in its existence. So today we learnt some very important things about derivatives, In order to get help on Is 1.5 a Rational Number and Is -16 an Irrational Number visit

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Rational numbers as exponents