# Trigonometry Differentiation Rules

A derivative of a function is the rate of change of the function or the slope of the line at a given point. The derivative of f(a) is notated as ${f}^{\prime }\left(a\right)$ or $\frac{d}{dx}f\left(a\right)$.

This discussion will focus on the basic trigonometric differentiation rules.

DERIVATIVES OF TRIGONOMETRIC FUNCTIONS:

 FUNCTION DERIVATIVE FUNCTION DERIVATIVE $\frac{d}{dx}\mathrm{sin}x$ cos x $\frac{d}{dx}\mathrm{csc}x$ -csc x cot x $\frac{d}{dx}\mathrm{cos}x$ -sin x $\frac{d}{dx}\mathrm{sec}x$ sec x tan x $\frac{d}{dx}\mathrm{tan}x$ sec2 x $\frac{d}{dx}\mathrm{cot}x$ -csc2 x

Let's look at some examples:

To work these examples requires the use of various differentiation rules. If you are not familiar with a rule go to the associated topic for a review.

20tan x
 Step 1: Apply the Constant Multiple Rule. $\frac{d}{dx}\left[cf\left(x\right)\right]=c\frac{d}{dx}f\left(x\right)$ $20\frac{d}{dx}\mathrm{tan}x$ Constant Mul. Step 2: Take the derivative of tan x. 20sec2 x Tangent Rule
Example 1:      sinx cosx
 Step 1: Apply the product rule. $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$ Step 2: Take the derivative of each part. Apply the appropriate trigonometric differentiation rule. $-\mathrm{sin}x$ Sine Rule __________________________ $\mathrm{cos}x$   Cosine Rule Step 3: Substitute the derivatives & simplify. $-{\mathrm{sin}}^{2}x+{\mathrm{cos}}^{2}x$
Example 2:
 Step 1: Apply the quotient rule. $\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]-f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]}{{\left[g\left(x\right)\right]}^{2}}$ $\frac{d}{dx}\left[\frac{1+\mathrm{sin}x}{\mathrm{cos}x}\right]$ Step 2: Take the derivative of each part. Apply the appropriate trigonometric differentiation rule. $\frac{d}{dx}1+\mathrm{sin}x$ Original $\frac{d}{dx}1+\frac{d}{dx}\mathrm{sin}x$       Sum Rule 0 + cos x Constant/Sine cos x __________________________ $\frac{d}{dx}\mathrm{cos}x$   Original -sin x       Sine Rule -sin x Step 3: Substitute the derivatives & simplify. $\frac{{\mathrm{cos}}^{2}x+\mathrm{sin}x+{\mathrm{sin}}^{2}x}{{\mathrm{cos}}^{2}x}$ $\frac{1+\mathrm{sin}x}{{\mathrm{cos}}^{2}x}$

 Related Links: Math algebra Inverse Trigonometric Differentiation Rules Logarithmic Differentiation Calculus Topics

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