Question

Differentiate the following functions. $$ u=\cos a x $$ $$ \frac{d u}{d x}=-a \sin a x $$

Short answer

Answer: The derivative of the function u = cos(ax) with respect to x is $$\frac{du}{dx} = -a\sin(ax)$$.

Step-by-step solution

Identify the Inner and Outer Functions

For the function u = cos(ax), we have: - Inner function: ax - Outer function: cos(u) We will differentiate the outer function and the inner function separately.

Differentiate the Inner Function

To differentiate the inner function ax with respect to x, we simply find the derivative of ax, which is: $$\frac{d(ax)}{dx} = a$$

Differentiate the Outer Function

To differentiate the outer function cos(u), we need to find the derivative of cos(u) with respect to u, which is: $$\frac{d(\cos(u))}{du}=-\sin(u)$$

Apply the Chain Rule

Now we can apply the chain rule, which states that the derivative of a composite function is the product of the derivatives of the inner and outer functions. In this case, we have: $$\frac{du}{dx} = \frac{d(\cos(ax))}{dx} = \frac{d(\cos(u))}{du} \cdot \frac{d(ax)}{dx}$$ Substitute the values we calculated in steps 2 and 3: $$\frac{du}{dx} = -\sin(u) \cdot a$$

Replace u with ax

Finally, we need to replace u with ax in our answer since we are looking for the derivative with respect to x: $$\frac{du}{dx} = -a\sin(ax)$$ Thus, the derivative of the function u = cos(ax) with respect to x is: $$\frac{du}{dx} = -a\sin(ax)$$

Key concepts

Chain Rule

In calculus, the chain rule is an essential formula for computing the derivative of composite functions. Composite functions are functions that are made up of other functions. An example is when you have a function inside another function, like in this exercise where we have \( u = \cos(ax) \). Here, \( ax \) is the inner function, and \( \cos(u) \) where \( u = ax \) is the outer function.

To use the chain rule, follow these steps:

• Identify the inner and outer functions: Determine which function is inside and which function encloses it.
• Differentiate each function separately: Find the derivative of the outer function with respect to its inner function, and the derivative of the inner function with respect to the variable.
• Multiply the derivatives: The chain rule states that the derivative of the composite function is the product of these two derivatives.

In this exercise, applying the chain rule results in \( \frac{du}{dx} = -a\sin(ax) \). This shows how the chain rule helps to differentiate complex functions by breaking them into simpler parts.

Derivatives

Derivatives are a fundamental concept in calculus and measure how a function changes as its input changes. In simpler terms, the derivative tells us the rate of change or the slope of the function at a given point. In this exercise, we are differentiating a trigonometric function involving an additional inner function, \( ax \).

Finding derivatives involves different rules depending on the function type, such as the power rule, product rule, and as seen here, the chain rule. For the inner function \( ax \), the derivative is straightforward:

• Differentiate \( ax \) with respect to \( x \): \( \frac{d(ax)}{dx} = a \).

The outer function \( \cos(ax) \) involves differentiating a trigonometric function, which takes us to the next concept. Understanding derivatives allows us to solve real-world problems involving motion, growth, and more by analyzing how quantities change.

Trigonometric Functions

Trigonometric functions like sine and cosine are essential in mathematics and are used to describe angles and periodic phenomena. When calculating derivatives of trigonometric functions, specific rules apply, as they behave differently from polynomial functions.

For cosine functions, the derivative formula is crucial. The derivative of \( \cos(u) \) with respect to \( u \) is \( -\sin(u) \). It's important to remember, trigonometric functions often come up in physics and engineering, especially when dealing with waves and rotational motions.

• Derivative of Cosine: Remember that the derivative of \( \cos(u) \) is \( -\sin(u) \), reflecting how closely related cosine and sine are.

When we combine this result with the inner function's derivative using the chain rule, it helps find how the entire expression \( \cos(ax) \) changes with \( x \). Recognizing how trigonometric functions differentiate enables us to handle functions involving these shapes easily.