Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int^{x^{2}} \cot 3 t d t$$

Short answer

\(dy/dx = \cot 3(x^2) \cdot 2x\).

Step-by-step solution

Identify the Integral

The function to be differentiated is \(y=\int^{{x^{2}}} \cot 3 t d t\). The variable of integration is \(t\) and the upper limit of integration is \(x^2\).

Apply Leibniz's Rule

According to Leibniz's Rule, if we have an integral in the format \(\int^{g(x)}f(t)dt\), its derivative with respect to \(x\) is \(f(g(x))\cdot g'(x)\). In our case, \(f(t) = \cot 3t\) and \(g(x) = x^2\). Substituting these values, the derivative will be \(\cot 3(x^2)\cdot g'(x)\).

Compute the Derivative

Now we must determine the derivative of \(x^2\), which is \(g'(x) = 2x\). Substituting this into our equation from Step 2, we get \(dy/dx = \cot 3(x^2) \cdot 2x\).

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on the concept of integration, which, together with differentiation, forms the fundamental theorem of calculus. Integration is the process of finding the integral of a function, which represents the accumulation of quantities, such as areas under curves, total distance, or mass of an object.

For instance, if you want to calculate the area under a curve between two points on the x-axis, you would evaluate the definite integral of the function that describes the curve. This critical concept is not just theoretical; it has real-world applications in physics, engineering, economics, and beyond.

Returning to our exercise, the expression \(y=\int^{x^{2}} \cot 3t \, dt\) involves finding the integral of the cotangent function with a variable upper limit, which is the square of \(x\). Understanding how to evaluate such integrals is a foundational skill in integral calculus.

Differentiation of Integrals

The differentiation of integrals is the operation of finding the derivative of an integral with respect to one of its limits, assuming that limit is a function of a certain variable. This operation is based on Leibniz's rule, which provides a method to differentiate under the integral sign when the limits of integration are functions of the differentiation variable.

Applying this rule simplifies the process of finding the derivative in situations where the antiderivative of the integrand is complicated or unknown. For the integral \(y=\int^{x^{2}} \cot 3t \, dt\) from our exercise, Leibniz's rule is the tool that lets us find the derivative with respect to \(x\) without explicitly computing the antiderivative of \(\cot 3t\).

Thus, we can say that the differentiation of integrals is a form of 'shortcut', allowing us to bypass sometimes arduous algebra to reach a solution more directly and elegantly.

Chain Rule

The chain rule is a fundamental principle in calculus used to find the derivative of composite functions. It states that if you have a function \(h\) that is the composition of two functions \(f\) and \(g\), so that \(h(x) = f(g(x))\), then the derivative of \(h\text{ with respect to }\ x\) is given by \(h'(x) = f'(g(x))\cdot g'(x)\).

This tool becomes incredibly useful when combined with Leibniz's rule, as seen in our text problem. The integrand \( \cot 3t\) becomes a composite function after integration—depending on \(x^2\) as its argument. To find the derivative of \(y\), we apply the chain rule: the derivative of \(\cot 3(x^2)\) with respect to \(x^2\) is \(\cot 3(x^2)\), and \(x^2\)'s derivative with respect to \(x\) is \(2x\), subsequently multiplying the two results together.