Differentiate trigonometric functions The derivative of sin x \sin x sin x is cos x \cos x cos x , the derivative of cos x \cos x cos x is − sin x -\sin x − sin x , and the derivative of tan x \tan x tan x is sec 2 x \sec^2 x sec 2 x , with angles measured in radians; these rules express instantaneous rate of change and tangent slope for periodic functions. The understanding includes applying the product, quotient, and chain rules to basic trigonometric expressions, while inverse, hyperbolic, and more generalized trigonometric derivatives are beyond this scope.
Detailed Explanation: Differentiate trigonometric functions To differentiate trigonometric functions, use these basic rules, with angles measured in radians:
d d x ( sin x ) = cos x \frac{d}{dx}(\sin x)=\cos x d x d ( sin x ) = cos x d d x ( cos x ) = − sin x \frac{d}{dx}(\cos x)=-\sin x d x d ( cos x ) = − sin x d d x ( tan x ) = sec 2 x \frac{d}{dx}(\tan x)=\sec^2 x d x d ( tan x ) = sec 2 x If the trigonometric function contains something other than just x x x chain rule :
d d x ( sin ( u ) ) = cos ( u ) d u d x \frac{d}{dx}\bigl(\sin(u)\bigr)=\cos(u)\frac{du}{dx} d x d ( sin ( u ) ) = cos ( u ) d x d u For a product of two functions, use the product rule :
d d x ( f g ) = f ′ g + f g ′ \frac{d}{dx}(fg)=f'g+fg' d x d ( f g ) = f ′ g + f g ′ Example
Differentiate
y = x 2 sin ( 3 x ) . y=x^2\sin(3x). y = x 2 sin ( 3 x ) . Step 1: Identify the product
The expression is the product of
f ( x ) = x 2 and g ( x ) = sin ( 3 x ) . f(x)=x^2
\qquad\text{and}\qquad
g(x)=\sin(3x). f ( x ) = x 2 and g ( x ) = sin ( 3 x ) . So we use the product rule:
y ′ = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) . y'=f'(x)g(x)+f(x)g'(x). y ′ = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) . Step 2: Differentiate each factor
For the first factor,
f ′ ( x ) = d d x ( x 2 ) = 2 x . f'(x)=\frac{d}{dx}(x^2)=2x. f ′ ( x ) = d x d ( x 2 ) = 2 x . For the second factor, use the chain rule. The outside function is sin \sin sin 3 x 3x 3 x
d d x ( sin ( 3 x ) ) = cos ( 3 x ) ⋅ d d x ( 3 x ) = 3 cos ( 3 x ) . \frac{d}{dx}\bigl(\sin(3x)\bigr)
=\cos(3x)\cdot\frac{d}{dx}(3x)
=3\cos(3x). d x d ( sin ( 3 x ) ) = cos ( 3 x ) ⋅ d x d ( 3 x ) = 3 cos ( 3 x ) . Thus,
g ′ ( x ) = 3 cos ( 3 x ) . g'(x)=3\cos(3x). g ′ ( x ) = 3 cos ( 3 x ) . Step 3: Substitute into the product rule
y ′ = ( 2 x ) sin ( 3 x ) + x 2 ( 3 cos ( 3 x ) ) . y'=(2x)\sin(3x)+x^2\bigl(3\cos(3x)\bigr). y ′ = ( 2 x ) sin ( 3 x ) + x 2 ( 3 cos ( 3 x ) ) . Therefore,
y ′ = 2 x sin ( 3 x ) + 3 x 2 cos ( 3 x ) . \boxed{y'=2x\sin(3x)+3x^2\cos(3x)}. y ′ = 2 x sin ( 3 x ) + 3 x 2 cos ( 3 x ) . The derivative gives the slope of the tangent to the curve at any value of x x x