2WBB0Calculus I

2WBB0 · Topic 05

Differentiation, implicit curves, and MVT

Derivative rules, tangent and normal lines, inverse derivatives, and short theorem proofs.

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What you need to be able to do

  • Differentiate nested products and quotients without losing chain factors.
  • Find tangent and normal lines from implicit equations.
  • Write a complete Mean Value Theorem inequality proof.
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1. Product, quotient, and chain rules

Most derivative errors are bookkeeping errors. Name the outer and inner functions or the numerator and denominator before differentiating.

Logarithmic and exponential derivatives often combine several rules. Keep the original inner expression visible until its derivative has been multiplied.

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Exam method

  • Mark each product, quotient, and composition before starting.
  • Differentiate one structural layer at a time.
  • Factor the final answer only after the derivative is correct.
  • Check dimensions and signs with a simple input value if possible.
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Product, quotient, and chain rules: worked examples

2 questions

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Product, quotient, and chain rulesSelf-review

Worked example 1

Differentiate h(x)=ln⁡(ln⁡x).h(x)=\ln(\ln x).
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2. Implicit differentiation and tangent or normal lines

When x and y share an equation, treat y as a function of x. Every derivative of an expression containing y receives a y′ factor.

After finding y′, evaluate it at the given point. A tangent uses that slope; a normal uses its negative reciprocal, with separate handling for zero or vertical slopes.

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Exam method

  • Differentiate both sides term by term with respect to x.
  • Apply product rule to xy terms and chain rule to powers of y.
  • Collect all y′ terms and solve for y′.
  • Evaluate at the point, choose tangent or normal slope, and write point-slope form.
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Implicit differentiation and tangent or normal lines: worked examples

2 questions

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Implicit differentiation and tangent or normal linesSelf-review

Worked example 1

Find the normal line to y2+3xy+x2=20y^2+3xy+x^2=20 at (2,2)(2,2).
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3. Mean Value Theorem proofs

A full Mean Value Theorem proof names the function and interval, checks continuity and differentiability, states the theorem equation, then bounds the derivative.

The theorem converts a difference of function values into a derivative at an unknown interior point. This is exactly what an inequality proof needs.

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Exam method

  • Choose f and the closed interval between the two inputs.
  • State that f is continuous on the closed interval and differentiable inside it.
  • Apply the theorem to obtain a point c.
  • Take absolute values or use derivative bounds to reach the requested inequality.
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Mean Value Theorem proofs: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

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Mean Value Theorem proofsSelf-review

Worked example 1

Prove ∣cos⁡a−cos⁡b∣≤∣a−b∣.|\cos a-\cos b|\le|a-b|.
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Formulas used in this topic

  • (uv)′=u′v+uv′,(uv)′=u′v−uv′v2(uv)'=u'v+uv',\qquad \left(\frac uv\right)'=\frac{u'v-uv'}{v^2}
  • ddxf(g(x))=f′(g(x))g′(x)\frac d{dx}f(g(x))=f'(g(x))g'(x)
  • y−y0=m(x−x0),mnormal=−1/mtangenty-y_0=m(x-x_0),\qquad m_{\rm normal}=-1/m_{\rm tangent}
  • f(b)−f(a)=f′(c)(b−a),c∈(a,b)f(b)-f(a)=f'(c)(b-a),\qquad c\in(a,b)
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