MATH 12002 - CALCULUS I 2.3 & 2.4: Derivatives of Trigonometric - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I § 2.3 & § 2.4: Derivatives of Trigonometric Functions Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 / 11

Derivatives of Sine & Cosine Our first two differentiation formulas for the trigonometric functions are d d dx sin x = cos x and dx cos x = − sin x . The first of these is proved in the text; we will prove the second using the definition of derivative. We will also need the angle sum formula for cosine, cos( A + B ) = cos A cos B − sin A sin B , and the limits from Equation 6 and Example 11 of § 1.4, sin θ cos θ − 1 lim = 1 and lim = 0 . θ θ θ → 0 θ → 0 D.L. White (Kent State University) 2 / 11

Derivatives of Sine & Cosine By definition of the derivative, we have d cos( x + h ) − cos x dx cos x = lim h h → 0 [cos x cos h − sin x sin h ] − cos x = lim h h → 0 cos x (cos h − 1) − sin x sin h = lim h h → 0 � � cos x cos h − 1 − sin x sin h = lim h h h → 0 � cos x cos h − 1 � � sin x sin h � = lim − lim h h h → 0 h → 0 � cos h − 1 � � sin h � = (cos x ) lim − (sin x ) lim h h h → 0 h → 0 = (cos x )(0) − (sin x )(1) = − sin x . D.L. White (Kent State University) 3 / 11

Derivatives of Other Trigonometric Functions Recall that the other trigonometric functions can be written in terms of sin x and cos x : tan x = sin x cot x = cos x cos x , sin x , 1 1 sec x = csc x = cos x , sin x . We can use these relations and the derivatives of sin x and cos x to find the derivatives of all of the trigonometric functions. D.L. White (Kent State University) 4 / 11

Derivatives of Other Trigonometric Functions Using the Quotient Rule, we have � sin x � d d dx tan x = cos x dx � d � d � � dx (sin x ) cos x − sin x dx (cos x ) = cos 2 x (cos x ) cos x − sin x ( − sin x ) = cos 2 x cos 2 x + sin 2 x 1 cos 2 x = sec 2 x . = = cos 2 x dx tan x = sec 2 x . Similarly, dx cot x = − csc 2 x . d d Hence D.L. White (Kent State University) 5 / 11

Derivatives of Other Trigonometric Functions Again using the Quotient Rule, we have d d � 1 � dx sec x = dx cos x � d � d � � dx (1) cos x − 1 · dx (cos x ) = cos 2 x (0) cos x − 1( − sin x ) = cos 2 x sin x cos x · sin x 1 = cos 2 x = cos x = sec x tan x . d d Hence dx sec x = sec x tan x . Similarly, dx csc x = − csc x cot x . D.L. White (Kent State University) 6 / 11

Derivatives of Other Trigonometric Functions We now have Derivatives of Trigonometric Functions d d dx sin x = cos x dx cos x = − sin x dx tan x = sec 2 x dx cot x = − csc 2 x d d d d dx sec x = sec x tan x dx csc x = − csc x cot x Notes: Derivatives involving cot x and csc x may show up in homework problems on WebAssign occasionally due to randomization, but they will not appear on any exams. The derivative of sec x is the product , sec x tan x = (sec x )(tan x ), and not the composite function, sec tan x = sec(tan x ). D.L. White (Kent State University) 7 / 11

Examples 3 sin x 1 Find the derivative of f ( x ) = √ x + 5 x 2 . By the quotient rule ( √ x + 5 x 2 ) − (3 sin x ) dx ( √ x + 5 x 2 ) � d � d � � dx (3 sin x ) f ′ ( x ) = ( √ x + 5 x 2 ) 2 (3 cos x )( √ x + 5 x 2 ) − (3 sin x )( 1 2 x − 1 / 2 + 10 x ) = ( √ x + 5 x 2 ) 2 . D.L. White (Kent State University) 8 / 11

Examples 2 Find the derivative of F ( x ) = 3 x 5 tan x . By the product rule, with f ( x ) = 3 x 5 and g ( x ) = tan x , � d � d � � dx (3 x 5 ) (tan x ) + (3 x 5 ) F ′ ( x ) = dx (tan x ) 15 x 4 tan x + 3 x 5 sec 2 x . = D.L. White (Kent State University) 9 / 11

Examples 3 Find the derivative of F ( x ) = x 2 sin x cos x . By the general product rule, with f ( x ) = x 2 , g ( x ) = sin x , and h ( x ) = cos x , � d sin x cos x + x 2 � d � d cos x + x 2 sin x dx x 2 � F ′ ( x ) � � = dx sin x dx cos x 2 x sin x cos x + x 2 cos x cos x + x 2 sin x ( − sin x ) . = D.L. White (Kent State University) 10 / 11

Examples 4 Find the derivative of f ( x ) = sec x tan x . cos x By the quotient rule � d � d � � dx (sec x tan x ) (cos x ) − (sec x tan x ) dx (cos x ) f ′ ( x ) = cos 2 x (sec x tan x ) tan x + sec x (sec 2 x ) � � cos x − (sec x tan x )( − sin x ) = . cos 2 x D.L. White (Kent State University) 11 / 11