5.5 Contents

1. $d/d\theta\sin(\theta)$

2. $d/d\theta\cos(\theta)$

3. $d/d\theta\tan(\theta)$

4. $d/d\theta\cot(\theta)$

5. $d/d\theta\sec(\theta)$ 🔧

6. $d/d\theta\csc(\theta)$ 🔧

7. $d/dx \sin^{-1}(x)$ 🔧

8. $d/dx \cos^{-1}(x)$ 🔧

9. $d/dx \tan^{-1}(x)$ 🔧

1. $d/dx \cot^{-1}(x)$ 🔧

2. $d/dx \sec^{-1}(x)$ 🔧

3. $d/dx \csc^{-1}(x)$ 🔧

5.5.1 $d/d\theta\sin(\theta)$

$$\frac{d}{d\theta}\sin(\theta)=\cos(\theta)$$

Proof

Insert $\sin(\theta)$ into the derivative function $$\lim_{\Delta\theta→0}\frac{\sin(\theta+\Delta\theta)-sin(\theta)}{\Delta\theta}$$ Expand the sine addition $$\lim_{\Delta\theta→0}\frac{\sin(\Delta\theta)\cdot\cos(\theta)+\sin(\theta)\cdot\cos(\Delta\theta)-sin(\theta)}{\Delta\theta}$$ Factor into terms of limits of trig functions $$\lim_{\Delta\theta→0}\bigg(\frac{\sin(\Delta\theta)}{\Delta\theta}\cdot\cos(\theta)+\frac{\cos(\Delta\theta)-1}{\Delta\theta}\cdot\sin(\theta)\bigg)$$ Solve the limits $$1\cdot\cos(\theta)+0\cdot\sin(\theta)$$ Simplify

5.5.2 $d/d\theta\cos(\theta)$

$$\frac{d}{d\theta}\cos(\theta)=-\sin(\theta)$$

Proof

Insert $\cos(\theta)$ into the derivative function $$\lim_{\Delta\theta→0}\frac{\cos(\theta+\Delta\theta)-cos(\theta)}{\Delta\theta}$$ Expand the cosine addition $$\lim_{\Delta\theta→0}\frac{\cos(\Delta\theta)\cdot\cos(\theta)-\sin(\theta)\cdot\sin(\Delta\theta)-cos(\theta)}{\Delta\theta}$$ Factor into terms of limits of trig functions $$\lim_{\Delta \theta→0}\bigg(\frac{\cos(\Delta\theta)-1}{\Delta\theta}\cdot\cos(\theta)-\frac{\sin(\Delta\theta)}{\Delta\theta}\cdot\sin(\theta)\bigg)$$ Solve the limits $$0\cdot\cos(\theta)-1\cdot\sin(\theta)$$ Simplify

5.5.3 $d/d\theta\tan(\theta)$

$$\frac{d}{d\theta}\tan(\theta)=\sec^2(\theta)$$

Proof

Write in terms of the tangent cofunctions $$\frac{d}{d\theta}\frac{\sin(\theta)}{\cos(\theta)}$$ Apply the quotient rule $$\frac{d}{d\theta}\frac{\cos(\theta)\cdot\sin(\theta)'-\sin(\theta)\cdot\cos(\theta)'}{\cos^2(\theta)}$$ Derive $$\frac{\cos^2(\theta)+\sin^2(\theta)}{\cos^2(\theta)}$$ Substitute the right angle identity $$1/\cos^2(\theta)$$ Write in terms of its cofunction

5.5.4 $d/d\theta\cot(\theta)$

$$\frac{d}{d\theta}\cot(\theta)=-\csc^2(\theta)$$

Proof

Write in terms of the cotangent cofunctions $$\frac{d}{d\theta}\frac{\cos(\theta)}{\sin(\theta)}$$ Apply the quotient rule $$\frac{d}{d\theta}\frac{\sin(\theta)\cdot\cos(\theta)'-\cos(\theta)\cdot\sin(\theta)'}{\sin^2(\theta)}$$ Derive $$-\frac{\sin^2(\theta)+\cos^2(\theta)}{\sin^2(\theta)}$$ Substitute the right angle identity $$-1/\sin^2(\theta)$$ Write in terms of its cofunction

5.5.5 $d/d\theta\sec(\theta)$ 🔧

$$\frac{d}{d\theta}\sec(\theta)=\sec(\theta)\cdot\tan(\theta),\medspace\cos(\theta)\ne 0$$

Proof

Write in terms of the cosine cofunction $$\frac{d}{dx}\frac{1}{\cos(\theta)}$$ Apply the quotient rule $$\frac{d}{dx}\frac{1}{\cos(\theta)}=\frac{\cos(\theta)\cdot 0-1\cdot\big(-\sin(\theta)\big)}{\cos^2(\theta)}$$ Simplify $$\frac{d}{dx}\frac{1}{\cos(\theta)}=\frac{\sin(\theta)}{\cos^2(\theta)}$$ Substitute its cofunctions

5.5.6 $d/d\theta\csc(\theta)$ 🔧

$$\frac{d}{d\theta}\csc(\theta)=-\csc(\theta)\cdot\cot(\theta),\medspace\sin(\theta)\ne 0$$

Proof

Write in terms of the sine cofunction $$\frac{d}{dx}\frac{1}{\sin(\theta)}$$ Apply the quotient rule $$\frac{d}{dx}\frac{1}{\sin(\theta)}=\frac{\sin(\theta)\cdot 0-1\cdot\cos(\theta)}{\sin^2(\theta)}$$ Simplify $$\frac{d}{dx}\frac{1}{\sin(\theta)}=-\frac{\cos(\theta)}{\sin^2(\theta)}$$ Substitute its cofunctions

5.5.7 $d/dx \sin^{-1}(x)$ 🔧

$$\frac{d}{dx}\sin^{-1}\bigg(\pm\frac{x}{a}\bigg)=\pm\frac{1}{\sqrt{a^2-x^2}},\medspace |x| \lt 1$$

Proof

Let $y=\sin^{-1}(\pm x/a)$ so that $\sin(y)=\pm x/a$, then derive $$\frac{d}{dx}\sin(y)=\pm\frac{1}{a}\cdot\frac{d}{dx}x$$ Apply the chain rule on the left $$\cos(y)\cdot\frac{dy}{dx}=\pm\frac{1}{a}\cdot\frac{d}{dx}x$$ Evaluate on the right $$\cos(y)\cdot\frac{dy}{dx}=\pm\frac{1}{a}$$ Divide by $\cos(y)$ $$\frac{dy}{dx}=\pm\frac{1}{a\cdot\cos(y)}$$ Substitute the right angle identity for cosine $$\frac{dy}{dx}=\pm\frac{1}{a\cdot\sqrt{1-\sin^2(y)}}$$ Substitute $y$ and $\sin(y)$ for their given values $$\frac{d}{dx}\sin^{-1}\bigg(\pm \frac{x}{a}\bigg)=\pm\frac{1}{a\cdot\sqrt{1-x^2/a^2}}$$ Distribute $a$ into the radicand

5.5.8 $d/dx \cos^{-1}(x)$ 🔧

$$\frac{d}{dx}\cos^{-1}\bigg(\pm\frac{x}{a}\bigg)=\mp\frac{1}{\sqrt{a^2-x^2}},\medspace |x| \lt 1$$

Proof

Let $y=\cos^{-1}(\pm x/a)$ so that $\cos(y)=\pm x/a$, then derive $$\frac{d}{dx}\cos(y)=\pm\frac{1}{a}\cdot\frac{d}{dx}x$$ Apply the chain rule on the left $$-\sin(y)\cdot\frac{dy}{dx}=\pm\frac{1}{a}\cdot\frac{d}{dx}x$$ Evaluate on the right $$-\sin(y)\cdot\frac{dy}{dx}=\pm\frac{1}{a}$$ Divide by $-\sin(y)$ $$\frac{dy}{dx}=\mp\frac{1}{a\cdot\sin(y)}$$ Substitute the right angle identity for sine $$\frac{dy}{dx}=\mp\frac{1}{a\cdot\sqrt{1-\cos^2(y)}}$$ Substitute $y$ and $\cos(y)$ for their given values $$\frac{d}{dx}\cos^{-1}\bigg(\pm \frac{x}{a}\bigg)=\mp\frac{1}{a\cdot\sqrt{1-x^2/a^2}}$$ Distribute $a$ into the radicand

5.5.9 $d/dx \tan^{-1}(x)$ 🔧

$$\frac{d}{dx}\tan^{-1}\bigg(\pm\frac{x}{a}\bigg)=\pm\frac{a}{x^2+a^2}$$

Proof

5.5.A $d/dx \cot^{-1}(x)$ 🔧

$$\frac{d}{dx}\cot^{-1}\bigg(\pm\frac{x}{a}\bigg)=\mp\frac{a}{x^2+a^2}$$

Proof

5.5.B $d/dx \sec^{-1}(x)$ 🔧

$$\frac{d}{dx}\sec^{-1}\bigg(\pm\frac{x}{a}\bigg)=\pm\frac{a}{\sqrt{x^2\cdot(x^2-a^2)}},\medspace |x| \gt 1$$

Proof

5.5.C $d/dx \csc^{-1}(x)$ 🔧

$$\frac{d}{dx}\csc^{-1}\bigg(\pm\frac{x}{a}\bigg)=\mp\frac{a}{\sqrt{x^2\cdot(x^2-a^2)}},\medspace |x| \gt 1$$

Proof