Trigonometric derivatives are used to find the rate of change of functions involving sine, cosine, and other trig functions. This page focuses on basic sine and cosine derivative rules, along with product rule, quotient rule, and chain rule variations. These examples help build fluency with common Calculus 1 trig derivative problems.

Notes

 

Problems

Find the derivative of the function.

\(\textbf{1)}\) \(f(x)=-3 \cos ⁡x\)

Show Derivative

The derivative is \(f'(x)=3 \sin ⁡x\)

Show Work

\(\,\,\,\,\,f(x)=-3\cos x\)

\(\,\,\,\,\,\frac{d}{dx}\left(\cos x\right)=-\sin x\)

\(\,\,\,\,\,f'(x)=-3(-\sin x)\)

\(\,\,\,\,\,f'(x)=3\sin x\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=3 \sin ⁡x\)

 

\(\textbf{2)}\) \(f(x)=\pi-\cos ⁡x\)

Show Derivative

The derivative is \(f'(x)=\sin ⁡x\)

Show Work

\(\,\,\,\,\,f(x)=\pi-\cos x\)

\(\,\,\,\,\,\frac{d}{dx}\left(\pi\right)=0\)

\(\,\,\,\,\,\frac{d}{dx}\left(-\cos x\right)=\sin x\)

\(\,\,\,\,\,f'(x)=\sin x\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=\sin ⁡x\)

 

\(\textbf{3)}\) \(f(x) = \pi \sin x + 2\pi x\)

Show Derivative

The derivative is \(f'(x) = \pi \cos x + 2\pi\)

Show Work

\(\,\,\,\,\,f(x) = \pi \sin x + 2\pi x\)

\(\,\,\,\,\,\frac{d}{dx}\left(\sin x\right)=\cos x\)

\(\,\,\,\,\,\frac{d}{dx}\left(x\right)=1\)

\(\,\,\,\,\,f'(x) = \pi \cdot \cos x + 2\pi(1)\)

\(\,\,\,\,\,f'(x) = \pi \cos x + 2\pi\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = \pi \cos x + 2\pi\)

 

\(\textbf{4)}\) \(f(x)=-4 \sin⁡ x+4 \cos ⁡x\)

Show Derivative

The derivative is \(f'(x)=-4 \cos ⁡x-4 \sin ⁡x\)

Show Work

\(\,\,\,\,\,f(x) = -4 \sin x + 4 \cos x\)

\(\,\,\,\,\,\frac{d}{dx}\left(\sin x\right)=\cos x\)

\(\,\,\,\,\,\frac{d}{dx}\left(\cos x\right)=-\sin x\)

\(\,\,\,\,\,f'(x) = -4 \cdot \cos x + 4 \cdot (-\sin x)\)

\(\,\,\,\,\,f'(x) = -4 \cos x – 4 \sin x\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=-4 \cos ⁡x-4 \sin ⁡x\)

 

\(\textbf{5)}\) \(f(x)=5x^{3} \cos x\)

Show Derivative

The derivative is \(f'(x)=-5x^{3} \sin x+15x^{2} \cos x\)

Show Work

\(\,\,\,\,\,\text{This will use product rule}\)

\(\,\,\,\,\,f(x) = 5x^{3} \cos x\)

\(\,\,\,\,\,f'(x) = (5x^{3}) \cdot (\cos x)’ + (5x^{3})’ \cdot (\cos x)\)

\(\,\,\,\,\,f'(x) = 5x^{3} \cdot (-\sin x) + 15x^{2} \cdot \cos x\)

\(\,\,\,\,\,f'(x) = -5x^{3} \sin x + 15x^{2} \cos x\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=-5x^{3} \sin x+15x^{2} \cos x\)

 

\(\textbf{6)}\) \(f(x) = 4 x^2 \sin (x) \)

Show Derivative

The derivative is \(f'(x) = 4x^2 \cos x + 8x \sin x\)

Show Work

\(\,\,\,\,\,\text{This will use product rule}\)

\(\,\,\,\,\,f(x) = 4x^2 \sin x\)

\(\,\,\,\,\,f'(x) = 4x^2(\sin x)’ + (4x^2)’\sin x\)

\(\,\,\,\,\,f'(x) = 4x^2 \cos x + 8x \sin x\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = 4x^2 \cos x + 8x \sin x\)

 

\(\textbf{7)}\) \(f(x) = \cos (x) \sin (x)\)

Show Derivative

The derivative is \(f'(x) = \cos^2 x – \sin^2 x\)

Show Work

\(\,\,\,\,\,\text{This will use product rule}\)

\(\,\,\,\,\,f(x) = \cos x \cdot \sin x\)

\(\,\,\,\,\,f'(x) = (\cos x) \cdot (\sin x)’ + (\sin x) \cdot (\cos x)’\)

\(\,\,\,\,\,f'(x) = \cos x \cos x – \sin x \sin x\)

\(\,\,\,\,\,f'(x) = \cos^2 x – \sin^2 x\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = \cos^2 x – \sin^2 x\)

 

\(\textbf{8)}\) \(f(x) = \sin(3x^2)\)

Show Derivative

The derivative is \(f'(x) = 6x \cos(3x^2)\)

Show Work

\(\,\,\,\,\,\text{This will use chain rule}\)

\(\,\,\,\,\,f(x) = \sin(3x^2)\)

\(\,\,\,\,\,f'(x) = \cos(3x^2) \cdot (3x^2)’\)

\(\,\,\,\,\,f'(x) = \cos(3x^2) \cdot 6x\)

\(\,\,\,\,\,f'(x) = 6x \cos(3x^2)\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = 6x \cos(3x^2)\)

 

\(\textbf{9)}\) \(f(x) = \displaystyle \frac{\sin x}{x^2}\)

Show Derivative

The derivative is \(f'(x) = \frac{x \cos x – 2 \sin x}{x^3}\)

Show Work

\(\,\,\,\,\,\text{This will use quotient rule}\)

\(\,\,\,\,\,f(x) = \frac{\sin x}{x^2}\)

\(\,\,\,\,\,f'(x) = \frac{(\sin x)’ \cdot x^2 – \sin x \cdot (x^2)’}{(x^2)^2}\)

\(\,\,\,\,\,f'(x) = \frac{x^2 \cos x – 2 x \sin x}{x^4}\)

\(\,\,\,\,\,f'(x) = \frac{x\left(x \cos x – 2 \sin x\right)}{x^4}\)

\(\,\,\,\,\,f'(x) = \frac{x \cos x – 2 \sin x}{x^3}\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = \frac{x \cos x – 2 \sin x}{x^3}\)

 

\(\textbf{10)}\) \(f(x) = x^3 \sin(x^2)\)

Show Derivative

The derivative is \(f'(x) = 3x^2 \sin(x^2) + 2x^4 \cos(x^2)\)

Show Work

\(\,\,\,\,\,f(x) = x^3 \sin(x^2)\)

\(\,\,\,\,\,\text{This will use product rule and chain rule}\)

\(\,\,\,\,\,f'(x) = (x^3)’ \cdot \sin(x^2) + x^3 \cdot (\sin(x^2))’\)

\(\,\,\,\,\,f'(x) = 3x^2 \sin(x^2) + x^3 \cos(x^2) \cdot 2x\)

\(\,\,\,\,\,f'(x) = 3x^2 \sin(x^2) + 2x^4 \cos(x^2)\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = 3x^2 \sin(x^2) + 2x^4 \cos(x^2)\)

 

\(\textbf{11)}\) \(f(x) = \sin^2 x\)

Show Derivative

The derivative is \(f'(x) = 2 \sin x \cos x\)

Show Work

\(\,\,\,\,\,\text{This will use chain rule}\)

\(\,\,\,\,\,f(x) = \sin^2 x\)

\(\,\,\,\,\,f(x) = \left( \sin x \right)^2\)

\(\,\,\,\,\,f'(x) = 2 \sin x \cdot \left(\sin x \right)’\)

\(\,\,\,\,\,f'(x) = 2 \sin x \cdot \cos x\)

\(\,\,\,\,\,f'(x) = 2 \sin x \cos x\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = 2 \sin x \cos x\)

 

\(\textbf{12)}\) \(f(x) = x^2 \cos(2x)\)

Show Derivative

The derivative is \(f'(x) = 2x \cos(2x) – 2x^2 \sin(2x)\)

Show Work

\(\,\,\,\,\,\text{This will use product rule and chain rule}\)

\(\,\,\,\,\,f(x) = x^2 \cos(2x)\)

\(\,\,\,\,\,f'(x) = (x^2)’ \cdot \cos(2x) + x^2 \cdot (\cos(2x))’\)

\(\,\,\,\,\,f'(x) = 2x \cos(2x) + x^2 \cdot \left(-\sin(2x)\right) \cdot 2\)

\(\,\,\,\,\,f'(x) = 2x \cos(2x) – 2x^2 \sin(2x)\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = 2x \cos(2x) – 2x^2 \sin(2x)\)

 

\(\textbf{13)}\) \(f(x) = \displaystyle \frac{\cos x}{x}\)

Show Derivative

The derivative is \(f'(x) = \frac{-x \sin x – \cos x}{x^2}\)

Show Work

\(\,\,\,\,\,\text{This will use quotient rule}\)

\(\,\,\,\,\,f(x) = \frac{\cos x}{x}\)

\(\,\,\,\,\,f'(x) = \frac{(\cos x)’ \cdot x – \cos x \cdot (x)’}{x^2}\)

\(\,\,\,\,\,f'(x) = \frac{ \left(-\sin x \right) \cdot x – \cos x \cdot (1)}{(x)^2}\)

\(\,\,\,\,\,f'(x) = \frac{-x \sin x – \cos x}{x^2}\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = \frac{-x \sin x – \cos x}{x^2}\)

 

\(\textbf{14)}\) \(f(x) = x \sin(5x)\)

Show Derivative

The derivative is \(f'(x) = \sin(5x) + 5x \cos(5x)\)

Show Work

\(\,\,\,\,\,\text{This will use product rule}\)

\(\,\,\,\,\,f(x) = x \sin(5x)\)

\(\,\,\,\,\,f'(x) = (x)’ \cdot \sin(5x) + x \cdot (\sin(5x))’\)

\(\,\,\,\,\,f'(x) = (1) \cdot \sin(5x) + x \cdot (\cos(5x)) \cdot 5\)

\(\,\,\,\,\,f'(x) = \sin(5x) + 5x \cos(5x)\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = \sin(5x) + 5x \cos(5x)\)

 

\(\textbf{15)}\) \(f(x) = \displaystyle \frac{x^3}{\sin x}\)

Show Derivative

The derivative is \(f'(x) = \frac{3x^2 \sin x – x^3 \cos x}{\sin^2 x}\)

Show Work

\(\,\,\,\,\,\text{This will use quotient rule}\)

\(\,\,\,\,\,f(x) = \frac{x^3}{\sin x}\)

\(\,\,\,\,\,f'(x) = \frac{(x^3)’ \cdot \sin x – x^3 \cdot (\sin x)’}{(\sin x)^2}\)

\(\,\,\,\,\,f'(x) = \frac{3x^2 \sin x – x^3 \cos x}{\sin^2 x}\)

\(\,\,\,\,\,\)The derivative is \(f'(x) = \frac{3x^2 \sin x – x^3 \cos x}{\sin^2 x}\)

 

\(\textbf{16)}\) \(f(x)=7\sin x-9\cos x\)

Show Derivative

The derivative is \(f'(x)=7\cos x+9\sin x\)

Show Work

\(\,\,\,\,\,f(x)=7\sin x-9\cos x\)

\(\,\,\,\,\,\frac{d}{dx}\left(\sin x\right)=\cos x\)

\(\,\,\,\,\,\frac{d}{dx}\left(\cos x\right)=-\sin x\)

\(\,\,\,\,\,f'(x)=7\cos x-9(-\sin x)\)

\(\,\,\,\,\,f'(x)=7\cos x+9\sin x\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=7\cos x+9\sin x\)

 

\(\textbf{17)}\) \(f(x)=\cos(4x^3)\)

Show Derivative

The derivative is \(f'(x)=-12x^2\sin(4x^3)\)

Show Work

\(\,\,\,\,\,f(x)=\cos(4x^3)\)

\(\,\,\,\,\,\text{This will use chain rule}\)

\(\,\,\,\,\,u=4x^3\)

\(\,\,\,\,\,u’=12x^2\)

\(\,\,\,\,\,f'(x)=-\sin(4x^3)\cdot 12x^2\)

\(\,\,\,\,\,f'(x)=-12x^2\sin(4x^3)\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=-12x^2\sin(4x^3)\)

 

\(\textbf{18)}\) \(f(x)=\left(x^2+1\right)\cos x\)

Show Derivative

The derivative is \(f'(x)=2x\cos x-\left(x^2+1\right)\sin x\)

Show Work

\(\,\,\,\,\,f(x)=\left(x^2+1\right)\cos x\)

\(\,\,\,\,\,\text{This will use product rule}\)

\(\,\,\,\,\,f'(x)=\left(x^2+1\right)(\cos x)’+\left(x^2+1\right)’\cos x\)

\(\,\,\,\,\,f'(x)=\left(x^2+1\right)(-\sin x)+2x\cos x\)

\(\,\,\,\,\,f'(x)=2x\cos x-\left(x^2+1\right)\sin x\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=2x\cos x-\left(x^2+1\right)\sin x\)

 

\(\textbf{19)}\) \(f(x)=\displaystyle\frac{x^2}{\cos x}\)

Show Derivative

The derivative is \(f'(x)=\displaystyle\frac{2x\cos x+x^2\sin x}{\cos^2 x}\)

Show Work

\(\,\,\,\,\,f(x)=\displaystyle\frac{x^2}{\cos x}\)

\(\,\,\,\,\,\text{This will use quotient rule}\)

\(\,\,\,\,\,f'(x)=\displaystyle\frac{(x^2)’\cos x-x^2(\cos x)’}{\left(\cos x\right)^2}\)

\(\,\,\,\,\,f'(x)=\displaystyle\frac{2x\cos x-x^2(-\sin x)}{\cos^2 x}\)

\(\,\,\,\,\,f'(x)=\displaystyle\frac{2x\cos x+x^2\sin x}{\cos^2 x}\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=\displaystyle\frac{2x\cos x+x^2\sin x}{\cos^2 x}\)

 

\(\textbf{20)}\) \(f(x)=\sin x \cos(2x)\)

Show Derivative

The derivative is \(f'(x)=\cos x\cos(2x)-2\sin x\sin(2x)\)

Show Work

\(\,\,\,\,\,f(x)=\sin x \cos(2x)\)

\(\,\,\,\,\,\text{This will use product rule and chain rule}\)

\(\,\,\,\,\,f'(x)=\sin x\left(\cos(2x)\right)’+(\sin x)’\cos(2x)\)

\(\,\,\,\,\,f'(x)=\sin x\left(-2\sin(2x)\right)+\cos x\cos(2x)\)

\(\,\,\,\,\,f'(x)=\cos x\cos(2x)-2\sin x\sin(2x)\)

\(\,\,\,\,\,\)The derivative is \(f'(x)=\cos x\cos(2x)-2\sin x\sin(2x)\)

 

 

See Related Pages\(\)

\(\bullet\text{Trig Derivative Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)

\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)

\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)

\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)

\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)

\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)

\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)

\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)

\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)

\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)

\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)