Single Variable Calculus

Differentiation

01The Derivative as the Slope of the Tangent Line 02Secant Lines and the Limit Definition of the Derivative 03Differentiating 1/x Straight From the Definition 04Tangents to 1/x and the Triangle of Area 2 05Newton and Leibniz Notation, and the Power Rule Problem set0/10 Problem set 20/10 MIT problem set0/3 Practice∞

01The Derivative as an Instantaneous Rate of Change 02Rates of Change Without Time: Gradient and Sensitivity 03Easy Limits, 0/0, and One-Sided Limits 04Continuity, Jumps, and Removable Discontinuities 05Infinite Discontinuities: 1/x and Its Derivative 06Differentiable Implies Continuous Problem set0/10 Problem set 20/10 Practice∞

01Two Kinds of Formula, and the Derivative of Sine 02The Derivative of Cosine, and Two Limits at Zero 03Bow and Bowstring: Proving the Two Trig Limits 04Why the Trig Derivatives Hold Only in Radians 05A Geometric Proof: Sine as a Height on the Circle 06Finishing the Proof, and the Product and Quotient Rules Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

01Proving the Product Rule: Change One Factor at a Time 02The Quotient Rule, and the Power Rule for Negative Exponents 03The Chain Rule: Differentiating a Function of a Function 04Higher Derivatives: The Sine Cycle and Three Notations 05The nth Derivative of xⁿ Is n Factorial, by Induction Problem set0/10 Problem set 20/10 MIT problem set0/2 Practice∞

01The Power Rule for Rational Exponents 02The Slope of a Circle, Two Ways 03Implicit Differentiation of a Quartic Curve 04Inverse Functions and the Reflection Across y = x 05Derivatives of the Arctangent and Arcsine Problem set0/10 Problem set 20/10 MIT problem set0/1 Practice∞

Derivatives of the Arctangent and Arcsine

How does one right triangle turn cos^2(arctan x) into 1/(1 + x^2), and what makes the square root in the arcsine derivative positive?

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Your summary note

1. The derivative of $\tan y$ and implicit differentiation
  Derive $\frac{d}{dy}\tan y = \frac{1}{\cos^2 y} = \sec^2 y$ with the quotient rule, then differentiate $\tan y = x$ with respect to $x$ and solve for $y'$ .
2. The right triangle that simplifies $\cos^2(\arctan x)$
  Draw the triangle with side $x$ opposite $y$ , adjacent side $1$ and hypotenuse $\sqrt{1 + x^2}$ , read off $\cos y$ , and finish with $\frac{d}{dx}\arctan x = \frac{1}{1 + x^2}$ .
3. The derivative of $\arcsin x$ and its branch
  Differentiate $\sin y = x$ implicitly, rewrite $\cos y$ as $\sqrt{1 - x^2}$ , state $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1 - x^2}}$ , and record the branch choice behind the positive square root.
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