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.

Using [latex]h=0.01[/latex], we estimate the derivative of the sine function as follows:

By defining [latex]D(x)=\frac{\sin(x+0.01)-\sin x}{0.01}[/latex] and plotting this using a graphing tool, we observe an approximation to the derivative of [latex]\sin x[/latex].

Upon examination, [latex]D(x)[/latex] 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 [latex]\sin x[/latex] is indeed [latex]\cos x[/latex].

If we were to follow the same steps to approximate the derivative of the cosine function, we would find that

The figure below shows the relationship between the graph of [latex]f(x)= \sin x[/latex] and its derivative [latex]f^{\prime}(x)= \cos x[/latex]. Notice that at the points where [latex]f(x)= \sin x[/latex] has a horizontal tangent, its derivative [latex]f^{\prime}(x)= \cos x[/latex] takes on the value zero. We also see that where [latex]f(x)= \sin x[/latex] is increasing, [latex]f^{\prime}(x)= \cos x>0[/latex] and where [latex]f(x)= \sin x[/latex] is decreasing, [latex]f^{\prime}(x)= \cos x<0[/latex].

Derivative of the Exponential Function

The differentiation of exponential functions [latex]B(x)=b^x,[/latex] begins by confirming that [latex]b^x[/latex] is defined for all real numbers and is inherently continuous. We assume [latex]B^{\prime}(0)[/latex] exists and is positive. In this context, we delve deeper into proving that [latex]B(x)[/latex] is differentiable across its entire domain by making one key assumption: there exists a unique value [latex]b > 0[/latex] for which [latex]B^{\prime}(0)=1[/latex].

The graph of [latex]E(x)=e^x[/latex] is shown alongside the line [latex]y=x+1[/latex] in Figure 2, demonstrating that the tangent to [latex]E(x)=e^x[/latex] at [latex]x=0[/latex] has a slope of [latex]1[/latex]. This observation supports the hypothesis that the value of [latex]e[/latex] optimizes the slope at [latex]x=0[/latex] to exactly [latex]1[/latex].

Now that we understand the underlying behavior at [latex]x=0[/latex], let’s derive the general derivative formula for [latex]B(x)=b^x, \, b>0[/latex]. We start by applying the limit definition of the derivative:

Turning to [latex]B^{\prime}(x)[/latex], we obtain the following.

We see that on the basis of the assumption that [latex]B(x)=b^x[/latex] is differentiable at [latex]0, \, B(x)[/latex] is not only differentiable everywhere, but its derivative is

For [latex]E(x)=e^x, \, E^{\prime}(0)=1[/latex]. Thus, we have [latex]E^{\prime}(x)=e^x[/latex]. (The value of [latex]B^{\prime}(0)[/latex] for an arbitrary function of the form [latex]B(x)=b^x, \, b>0[/latex], will be derived later.)

Derivative of the Logarithmic Function

Now that we have the derivative of the natural exponential function, we can use implicit differentiation to find the derivative of its inverse, the natural logarithmic function.

Now that we can differentiate the natural logarithmic function, we can use this result to find the derivatives of [latex]y=\log_b x[/latex] and [latex]y=b^x[/latex] for [latex]b>0, \, b\ne 1[/latex].