Chain rule calculus cos2x2/19/2024 The chain rule is used to find the derivative of a composite function. Thus the equation of the tangent line to the given function y = (5 x 4 - 2) 3 applying the chain rule formula is y = 540x - 513 Hence substitute (1,27) in the equation of the tangent line, y = 540x + b, we get We need to find the equation of the tangent line. Therefore the equation of the tangent line in the slope-intercept form is y = mx+ b ⇒ 27 = 540x + b We need to evaluate the function and the derivative at the given point We know that the derivative of the function gives the slope of the line at the given point. Let us apply the chain rule to find the equation of the tangent line to the given function y = (5 x 4 - 2) 3 at x = 1. ![]() To find the rate of change of the average molecular speed,.To determine if a function is increasing or decreasing,.To find the position of an object that is moving to the right and left in a particular interval,.To calculate the rate of change of distance between two moving objects,.To find the time rate of change of the pressure,.This chain rule has broad applications in the fields of physics, chemistry, and engineering. We can assume the expression that is replacing "x" with "u" and applying the chain rule formula.Įxample : To find d/dx (sin 2x), assume that y = sin 2x and 2x = u. Chain Rule Formula 1:Įxample : To find the derivative of d/dx (sin 2x), express sin 2x = f(g(x)), where f(x) = sin x and g(x) = 2x. There are two forms of chain rule formula as shown below. On simplifying we get, cos x/sin x = cot x.Finally g'(x) = derivative of the outside function, leaving the inside alone × the derivative of the inside function = 1/sin x × cos x. ![]() The derivative of the inner function is cos x.The derivative of the outer function is 1/sin x.sin x is the inner function and ln(x) is the outer function.Step 6: Simplify the chain rule derivative.įor example: Consider a function: g(x) = ln(sin x).Step 5: Multiply the results from step 4 and step 5. ![]()
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