Derivative From First Principles

The Calculus Is Easy… The Algebra Is the Hard Part

This derivative problem looks like a standard product rule + chain rule question, but the real challenge is the algebra simplification at the end. To get the correct answer, you need strong factoring skills. Many calculus students know the derivative rules but lose points when the algebra gets messy. Can you simplify it correctly before the ...

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For the first one, the derivative and integral of e^x is just e^x. For the second one, limits and power rule are only the basics of the average calc class. For the third one, as I said, d/dx of e^x = e^x. e^x is the only function that has this property. For the last one, cos(60) and sin(30) both equal 1/2, and also tan(26.565) is also about 1/2. . . . . #calculusmemes #calculus #mathmemes

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Day 162 — 365 Problems for 365 Days AP Calculus derivative with logarithms (quotient rule): d/dx [ x / ln(x) ] This is a high-yield calculus derivative because it mixes a rational expression with a natural log. The cleanest path is the quotient rule, but it also reinforces the key fact that d/dx ln(x) = 1/x. This exact pattern shows up in optimization, related rates, and in simplifying harder derivatives. Project Mentor helps high school students with AP self-study plans, competition prep, and r

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🔥 Day 152 — 2026 MIT Integration Bee (Qualifying) Q18 365 Problems for 365 Days Evaluate: Integral of (sin^2(x)/x^2 − sin(2x)/x) dx This one is a “recognize the derivative” speed problem. The integrand looks messy, but it’s literally the derivative of a super simple expression if you think “quotient rule trig identity.” If you try to integrate directly, it’s pain. If you spot the structure, it’s instant. Hints: \t•\tLook for something like (sin^2 x)/x \t•\tDifferentiate it with the quotient rul

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