Question

Derivatives of integrals Simplify the following expressions. $$\frac{d}{d z} \int_{\sin z}^{10} \frac{d t}{t^{4}+1}$$

Short answer

Answer: The simplified expression for the derivative is \(-\frac{\cos z}{(\sin z)^4 + 1}\).

Step-by-step solution

Identify the Components

In the given expression: $$\frac{d}{d z} \int_{\sin z}^{10} \frac{d t}{t^{4}+1}$$ We can identify the components as: - \(a(z) = \sin z\) and \(a'(z) = \cos z\) - \(b(z) = 10\) and \(b'(z) = 0\) - \(f(t, z) = \frac{1}{t^4 + 1}\), which doesn't have a \(z\) component, so no derivative with respect to \(z\) is needed.

Apply the Leibniz Rule

Using the Leibniz rule, we have: $$\frac{d}{d z} \int_{\sin z}^{10} \frac{d t}{t^{4}+1} = f(10, z) \cdot b'(z) - f(\sin z, z) \cdot a'(z) $$ Now, we plug in the identified components: $$\frac{d}{d z} \int_{\sin z}^{10} \frac{d t}{t^{4}+1} = \frac{1}{10^4 + 1} \cdot 0 - \frac{1}{(\sin z)^4 + 1} \cdot \cos z$$

Simplify the Expression

Since \(0\) multiplied by any expression is equal to \(0\), we are left with: $$-\frac{1}{(\sin z)^4 + 1} \cdot \cos z$$ Therefore, the simplified expression for the derivative is: $$\frac{d}{d z} \int_{\sin z}^{10} \frac{d t}{t^{4}+1} = -\frac{\cos z}{(\sin z)^4 + 1}$$

Key concepts

Leibniz Rule

The Leibniz Rule is a fundamental principle in calculus that allows us to differentiate an integral whose limits are functions of another variable. This rule extends the concept of differentiation to more complex scenarios where the limits of integration are not constants but instead depend on a variable.
The rule states that if you have an integral of the form:

• \( \int_{a(z)}^{b(z)} f(t, z) \ dt \)

where the limits \(a(z)\) and \(b(z)\) are functions of \(z\), the derivative of this integral with respect to \(z\) can be expressed as:

• \( \frac{d}{dz} \int_{a(z)}^{b(z)} f(t, z) \ dt = f(b(z), z) \cdot b'(z) - f(a(z), z) \cdot a'(z) + \int_{a(z)}^{b(z)} \frac{\partial f(t, z)}{\partial z} \ dt\)

This formula comprises three parts:

• The evaluation of the function \(f\) at the upper limit \(b(z)\) multiplied by the derivative of \(b(z)\).
• A subtraction of the function \(f\) evaluated at the lower limit \(a(z)\), multiplied by the derivative of \(a(z)\).
• An integral correction term accounting for any \(z\)-dependence of the function \(f(t, z)\).

In cases where \(f(t,z)\) does not depend on \(z\), the correction term is zero.

Integral Calculus

Integral calculus is a central part of calculus focused on the concept of integration. It is the mathematical discipline that deals with summing continuous quantities and finding accumulated quantities..
When you integrate, you essentially compute the area under a curve or the total accumulation of a quantity given a rate of accumulation.

• Indefinite Integrals: These are integrals without specified limits and include an arbitrary constant \(C\). They represent a family of functions and are written as \(\int f(x) \, dx\).
• Definite Integrals: These have specified limits and represent a number that quantifies the accumulated value of a function over a specific interval, written as \(\int_{a}^{b} f(x) \, dx\).

In applied mathematics, integral calculus is used to solve physical problems involving quantities such as length, area, and volume.
It complements differential calculus, which concerns rates of change and slopes of curves.

Derivative of an Integral

The derivative of an integral might sound complex, but it represents how the accumulated total changes in response to changes in its limits. When addressed with fundamental theorems and rules, such as the Leibniz Rule, it becomes quite manageable.
This operation arises in both theory and application, particularly where the bounds of an integral are not constants but vary with another parameter, like \(z\) in our problem.
The process includes evaluating the behavior of the integrand at the boundaries and the influence of these boundaries on the total value of the integral.
In simple terms, when you derive an integral:

• You assess how the changes in the limits, expressed as functions, influence the integral.
• You determine if the integrand itself changes, in which case additional calculations might be necessary.

Applications of this concept are extensive, finding relevance in areas such as physics, engineering, and any field where dynamic systems are analyzed.