Derivatives of Trig Functions
Trigonometry can be intimidating for some students, but we’ll only be looking at the derivatives of sine, cosine, tangent, and other trigonometric functions. Just finding the slope of a tangent line, increasing the number of things we can take the derivative of, incorporating them with our products and our quotients. [Graph] To this point, we have considered derivatives only of polynomial functions. However, rational expressions and square roots can be expressed as x raised to a power, so they are within the same general category of derivatives as polynomials. Let’s consider the derivative of sine of x using trigonometric functions. The calculator will now graph the derivative of sine, and we will see that this is a very familiar function. d/dx sinx=cosx
So that tells us that the derivative of the sine function is the cosine function. We will also use a calculator to graph the derivative of cosine
[Graph]
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Now that we know that the derivative of sine is cosine, it’s natural to think that cosine’s derivative is sine. And you’d be almost correct.
[Graph]
As shown in the graph, the derivative of cosine actually is negative first, but sine is positive to start. However, notice that when cosine has a peak or a value, the derivative is zero. Every peak and valley on the derivative is at these points where cosine equals zero. Therefore, it’s actually not sine.
d/dx sinx=cosx d/dx cosx=-sinx Tangent x is actually the negative of sine x. We will not be using such an approach for the derivatives of other trig functions. Instead, we can develop them using our knowledge of sine and cosine d/dx tanx d/dx(sinx/cosx)=(cosx*cosx-sinx(-sinx))/(cos^2 x) =(cos^2 x+sin^2 x)/(cos^2 x) =1/(cos^ x)=secx=d/dx(tanx) Tangent can be expressed as sine divided by cosine. And taking the derivative of this is a quotient, so we’ll call that high, this low. Derivative rules never go away, so we have the derivative of sine equals cosine, minus the derivative of cosine x, which is negative sine x. All divided by cosine squared x that’s low squared cosine, which is cosine squared; sine times sine is sine squared. Negative and negative are positive We divide that by cosine squared, then we have cosine squared plus sine squared over cosine squared. When we plug in our value for x, that’s when we can solve for secant x, which is tangent. So the derivative tangent with respect to x is secant x. Times cosine times d/dx(secx) d/dx(1/cosx) (cosx*0-1*(-sinx))/cos^2 x (sinx*1)/(cosx*cosx) tanx*secx=d/dx(secx) We’ll do one more derivative of a trig function, and then you can complete the rest. We can express this equation in terms of the derivative of cosine. The derivative of cosine is -1 over cosine squared. So we can rewrite this equation as 1 over cosine x squared, or 2.7. For this time and this problem only, we will do the derivative as a quotient even though there is no variable in the numerator. The top part will be called high, the bottom low. Low times the derivative of high times derivative of 1 over cosine squared x equals 0 minus high times derivative of low. d/dx sinx=cosx d/dx cosx=-sinx d/dx tanx=sec^2 x d/dx cotx=-csc^2 x d/dx secx=secxtanx d/dxcscx=-cscxcotx *Memorize The derivative of secant x with respect to x is tangent x secant x. The derivative of cotangent x, the derivative of a cosecant x, and the derivative of cosecant x will be written out momentarily. Here are all six trigonometric derivatives, in one place.
