- Calculate the derivatives of sine and cosine functions, including second derivatives and beyond
- Determine the derivatives for basic trig functions like tangent, cotangent, secant, and cosecant
Derivatives of the Sine and Cosine Functions
Simple harmonic motion, a type of periodic motion where the restoring force is directly proportional to the displacement, is best described using trigonometric functions like sine and cosine. The behavior of these functions, particularly how they change over time, is crucial in understanding motion dynamics. The derivatives of sine and cosine functions help us compute velocity and acceleration at any point in the motion, linking theoretical physics closely with calculus.
We begin our exploration of the derivative for the sine function by using the limit definition to estimate its derivative.
For a function f(x),f(x), the derivative f′(x)f′(x) is defined as:
This allows us to approximate f′(x)f′(x) for small values of hh as:
Using h=0.01h=0.01, we estimate the derivative of the sine function as follows:
By defining D(x)=sin(x+0.01)−sinx0.01D(x)=sin(x+0.01)−sinx0.01 and plotting this using a graphing tool, we observe an approximation to the derivative of sinxsinx.

Upon examination, D(x)D(x) appears to be a close match to the graph of the cosine function. This graphical analysis provides a practical demonstration of the derivative, confirming that the derivative of sinxsinx is indeed cosxcosx.
If we were to follow the same steps to approximate the derivative of the cosine function, we would find that
derivatives of sinxsinx and cosxcosx
The derivative of the sine function sinxsinx is the cosine function cosxcosx.
The derivative of the cosine function cosxcosx is the negative sine function −sinx−sinx.
Proof
Because the proofs for ddx(sinx)=cosxddx(sinx)=cosx and ddx(cosx)=−sinxddx(cosx)=−sinx use similar techniques, we provide only the proof for ddx(sinx)=cosxddx(sinx)=cosx.
Before beginning, it is important to recall two important trigonometric limits:
The graphs of y=(sinh)hy=(sinh)h and y=(cosh−1)hy=(cosh−1)h are shown in Figure 2.

We also recall the following trigonometric identity for the sine of the sum of two angles:
Now that we have gathered all the necessary equations and identities, we proceed with the proof.
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The figure below shows the relationship between the graph of f(x)=sinxf(x)=sinx and its derivative f′(x)=cosx. Notice that at the points where f(x)=sinx has a horizontal tangent, its derivative f′(x)=cosx takes on the value zero. We also see that where f(x)=sinx is increasing, f′(x)=cosx>0 and where f(x)=sinx is decreasing, f′(x)=cosx<0.

Find the derivative of f(x)=5x3sinx.
Find the derivative of g(x)=cosx4x2.
A particle moves along a coordinate axis in such a way that its position at time t is given by s(t)=2sint−t for 0≤t≤2π. At what times is the particle at rest?