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- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/05%3A_Differentiation_-_Differentiation_Rules_and_Formulas/5.05%3A_Chain_RuleWe want to derive a rule for the derivative of a composite function of the form f∘g in terms of the derivatives of f and g. Earlier, you were asked if you find the effect of g on the derivative by com...We want to derive a rule for the derivative of a composite function of the form f∘g in terms of the derivatives of f and g. Earlier, you were asked if you find the effect of g on the derivative by comparing the derivative of f(x)=x 2 with the derivative of f(x)=(5x) 2 where g(x)=5x. The derivative of a function is the slope of the line tangent to the function at a given point on the graph.
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/05%3A_Differentiation_-_Differentiation_Rules_and_Formulas/5.04%3A_Derivatives_of_Trigonometric_FunctionsAll of the derivatives can be proved by the definition of the derivative, but the reciprocal functions can be found using a simpler method. First, see if you can identify any points where you know the...All of the derivatives can be proved by the definition of the derivative, but the reciprocal functions can be found using a simpler method. First, see if you can identify any points where you know the derivative of sinx and cosx: Each function has two places in the interval 0≤x≤2π where the tangent line has a slope of 0. The derivative of a function is the slope of the line tangent to the function at a given point on the graph.
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/07%3A_Differentiation_-_Increasing_and_Decreasing_Values_and_Extrema/7.02%3A_Absolute_versus_Local_ExtremaIf you realized that the vertex of the quadratic function is also the location where the slope of the tangent line, and therefore the derivative, equals 0 you made a great observation. \( \frac{dy}{dx...If you realized that the vertex of the quadratic function is also the location where the slope of the tangent line, and therefore the derivative, equals 0 you made a great observation. dydx=2ax+b=0 means that x=−b2a. A critical value (or point) is a value in a closed interval of the domain of the function where either the function derivative at the point is 0, or the derivative does not exist.
- https://k12.libretexts.org/Bookshelves/Mathematics/Calculus/05%3A_Differentiation_-_Differentiation_Rules_and_Formulas/5.01%3A_Constant_Identity_and_Power_RulesBetween the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort - often in your head! (The Power Rule) If...Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort - often in your head! (The Power Rule) If n is a positive integer, then for all real values of x ddx[xn]=nxn−1 The derivative of a function is the slope of the line tangent to the function at a given point on the graph.
