Calculus - The Derivative - Quick Guides

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Study GuideCalculus–The Derivative1.The Chain RuleUnderstanding Composite FunctionsSuppose a function is written as:This means:•(h(x)) is theinside function•(g(x)) is theoutside functionTo find the derivative of (f(x)), we:1.Differentiate theoutside function2.Multiply by the derivative of theinside functionUsing the chain rule:Because two functions are involved, we must considerboth derivatives.When There Are More Than Two FunctionsNow consider a function with three layers:Here:•(p(x)) is the innermost function•(n(x)) is in the middle•(m(x)) is the outermost function

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Study GuideApplying the chain rule gives:A helpful strategy is to work from theoutside to the inside, taking one derivative at a time andmultiplying as you go.Example 1:Find (f'(x)) if•Outside function: (u8)•Inside function: (3x2+ 5x-2)Example 2:Find (f'(x)) if•Outside function: (tan u)•Inside function: (sec x)Example 3:Find(dy/dx) ifThis means (y = [sin(3x-1)]3).

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Study GuideExample 4:Find (f'(2)) ifFirst rewrite the function:Differentiate:Now substitute (x = 2):Example 5:Find the slope of the tangent line toat the point ((-1,-32)).The slope of the tangent line is the derivative.Now substitute (x =-1):So, the slope of the tangent line at ((-1,-32)) is–160.Key Takeaways•Thechain ruleis used for composite functions.•Always multiply by the derivative of theinside function.

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Study Guide•Work from theoutside to the inside.•Each nested function adds another derivative step.Once you get comfortable identifying inside and outside functions, the chain rule becomes a veryreliable and powerful tool in calculus.2.Implicit DifferentiationIn calculus, not every equation involvingxandyis written in the formSometimes,x and y are mixed together in one equation, making it difficult—or even impossible—tosolve explicitly for (y). Even so, we often assume thaty depends on x, meaning (y) is still a functionof (x).This is whereimplicit differentiationbecomes useful.What Is Implicit Differentiation?Implicit differentiationis a technique that allows us to find(dy/dx)without solving the equation fory first.The key idea is this:•Differentiateboth sidesof the equation with respect to (x).•Whenever you differentiate a term involving (y), you must apply thechain rule.•That means every derivative of (y) becomes multiplied by(dy/dx).Why the Chain Rule Matters HereSince we are assuming that (y) is a function of (x), any expression involving (y) must be treatedcarefully. Differentiating (y) is not the same as differentiating a constant—it changes with (x). That’swhy(dy/dx)appears throughout implicit differentiation.

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Study GuideExample 1:Find(dy/dx)ifDifferentiate both sides with respect to (x):•Use theproduct ruleon both (x2y3) and (xy)•Apply thechain rulewhen differentiating terms with (y)After simplifying and collecting terms involving(dy/dx), we get:Solving for(dy/dx):Example 2:Find (y') ifDifferentiate both sides with respect to (x):Now group the (y') terms:Solve for (y'):

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Study GuideExample 3:Find (y') at the point ((-1, 1)) ifDifferentiate implicitly:Group the (y') terms:Solve for (y'):Now substitute (x =-1) and (y = 1):Example 4:Find the slope of the tangent line toat the point ((3,-4)).Differentiate implicitly:Solve for (y'):Now substitute the point ((3,-4)):

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Study GuideSo, the slope of the tangent line at ((3,-4)) is3/4.Key Takeaways•Useimplicit differentiationwhen you cannot easily solve for (y).•Differentiate both sides with respect to (x).•Apply thechain rulewhenever a term contains (y).•Collect all(dy/dx)terms and solve for it.•You can find slopes and tangent lines directly from implicit equations.3.Higher-Order DerivativesWhat Are Higher-Order Derivatives?If a function is written asthen its first derivative isBecause (f'(x)) is also a function, we can take its derivative too. This gives us:•Second derivative:•Third derivative:•And so on…These are calledhigher-order derivatives.

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Study GuideNotation TipUsing prime symbols (′, ″, ‴) works well at first, but it quickly becomes confusing. That’s whymathematicians often usenumerical notation, such asto represent thenth derivativeof a function. This keeps things neat and clear.Example 1:Find the first, second, and third derivatives ofStart by differentiating step by step:First derivative:Second derivative:Third derivative:Each derivative comes from differentiating the one before it.Example 2:Find the first, second, and third derivatives ofThis is a composite function, so we use thechain rule.First derivative:

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Study GuideSecond derivative:Third derivative:This example shows how trigonometric identities and the chain rule often appear in higher-orderderivatives.Example 3:Find (f(3)(4)) ifFirst, rewrite the function using exponents:Now differentiate step by step:First derivative:Second derivative:Third derivative:Now evaluate at (x = 4):

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