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Calculus - The Derivative

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    Study Guide Calculus – The Derivative 1. The Chain Rule Understanding Composite Functions Suppose a function is written as: This means: • (h(x)) is the inside function • (g(x)) is the outside function To find the derivative of (f(x)), we: 1. Differentiate the outside function 2. Multiply by the derivative of the inside function Using the chain rule: Because two functions are involved, we must consider both derivatives . When There Are More Than Two Functions Now 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 Guide Applying the chain rule gives: A helpful strategy is to work from the outside to the inside , taking one derivative at a time and multiplying as you go. Example 1 : Find (f'(x)) if • Outside function: (u 8 ) • Inside function: (3x 2 + 5x - 2) Example 2 : Find (f'(x)) if • Outside function: (tan u) • Inside function: (sec x) Example 3 : Find (dy/dx ) if This means (y = [sin(3x - 1)] 3 ).

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    Study Guide Example 4 : Find (f'(2)) if First rewrite the function: Differentiate: Now substitute (x = 2): Example 5 : Find the slope of the tangent line to at 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 • The chain rule is used for composite functions. • Always multiply by the derivative of the inside function .

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    Study Guide • Work from the outside to the inside . • Each nested function adds another derivative step. Once you get comfortable identifying inside and outside functions, the chain rule becomes a very reliable and powerful tool in calculus. 2. Implicit Differentiation In calculus, not every equation involving x and y is written in the form Sometimes, x and y are mixed together in one equation , making it difficult — or even impossible — to solve explicitly for (y). Even so, we often assume that y depends on x , meaning (y) is still a function of (x). This is where implicit differentiation becomes useful. What Is Implicit Differentiation? Implicit differentiation is a technique that allows us to find (dy/dx) without solving the equation for y first . The key idea is this: • Differentiate both sides of the equation with respect to (x). • Whenever you differentiate a term involving (y), you must apply the chain rule . • That means every derivative of (y) becomes multiplied by (dy/dx) . Why the Chain Rule Matters Here Since we are assuming that (y) is a function of (x), any expression involving (y) must be treated carefully. Differentiating (y) is not the same as differentiating a constant — it changes with (x). That’s why (dy/dx) appears throughout implicit differentiation.

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    Study Guide Example 1 : Find (dy/dx) if Differentiate both sides with respect to (x): • Use the product rule on both (x 2 y 3 ) and (xy) • Apply the chain rule when differentiating terms with (y) After simplifying and collecting terms involving (dy/dx) , we get: Solving for (dy/dx) : Example 2 : Find (y') if Differentiate both sides with respect to (x): Now group the (y') terms: Solve for (y'):

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

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    Study Guide So, the slope of the tangent line at ((3, - 4)) is 3/4. Key Takeaways • Use implicit differentiation when you cannot easily solve for (y). • Differentiate both sides with respect to (x). • Apply the chain rule whenever 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 Derivatives What Are Higher - Order Derivatives? If a function is written as then its first derivative is Because (f'(x)) is also a function, we can take its derivative too. This gives us: • Second derivative: • Third derivative: • And so on… These are called higher - order derivatives .

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    Study Guide Notation Tip Using prime symbols (′, ″, ‴) works well at first, but it quickly becomes confusing. That’s why mathematicians often use numerical notation , such as to represent the nth derivative of a function. This keeps things neat and clear. Example 1 : Find the first, second, and third derivatives of Start 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 of This is a composite function, so we use the chain rule . First derivative:

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    Study Guide Second derivative: Third derivative: This example shows how trigonometric identities and the chain rule often appear in higher - order derivatives. Example 3 : Find (f (3) (4)) if First, 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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