Trigonometry Differentiation Rules

Step 1: Apply the Constant Multiple Rule.

$\frac{d}{dx}\left[cf\left(x\right)\right]=c\frac{d}{dx}f\left(x\right)$

$20\frac{d}{dx}\tan x$ Constant Mul.

Step 2: Take the derivative of tan x.

20sec² x Tangent Rule

Step 1: Apply the product rule.

$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$

$\frac{d}{dx}\left[sin x cos x\right]$

$sin x\frac{d}{dx}cos x+\cos x\frac{d}{dx}sin x$

Step 2: Take the derivative of each part.

Apply the appropriate trigonometric differentiation rule.

$\frac{d}{dx}cos x$

$-\sin x$ Sine Rule

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$\frac{d}{dx}sin x$

$\cos x$   Cosine Rule

Step 3: Substitute the derivatives & simplify.

$sin x(-\sin x)+\left(cos x\right)\left(\cos x\right)$

$-{\sin }^{2}x+{\cos }^{2}x$

Step 1: Apply the quotient rule.

$\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]-f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]}{{\left[g\left(x\right)\right]}^2}$

$\frac{d}{dx}\left[\frac{1+\sin x}{\cos x}\right]$

$\frac{\cos x \frac{d}{dx}(1+\sin x)-(1+\sin x)\frac{d}{dx}\cos x}{{\cos }^{2}x}$

Step 2: Take the derivative of each part.

Apply the appropriate trigonometric differentiation rule.

$\frac{d}{dx}1+\sin x$ Original

$\frac{d}{dx}1+\frac{d}{dx}\sin x$       Sum Rule

0 + cos x Constant/Sine

cos x

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$\frac{d}{dx}\cos x$   Original

-sin x       Sine Rule

-sin x

Step 3: Substitute the derivatives & simplify.

$\frac{\cos x \left(\cos x\right)-(1+\sin x) (-\sin x)}{{\cos }^{2}x}$

$\frac{{\cos }^{2}x+\sin x+{\sin }^{2}x}{{\cos }^{2}x}$

$\frac{1+\sin x}{{\cos }^{2}x}$