Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{0}^{x^{2}} e^{t^{2}} d t$$

Short answer

The derivative of the function y with respect to x is \(y' = 2x \cdot e^{x^4}\).

Step-by-step solution

Identify the integral function

Here, the function \(y\) is given as the definite integral of \(e^{t^2}\) from \(0\) to \(x^2\): \(y = \int_{0}^{x^{2}} e^{t^{2}} dt\). We are to differentiate this function with respect to \(x\).

Apply Leibniz Rule

From Leibniz Rule, if \(y = \int_{a}^{u(x)} f(t) dt\) where \(u(x)\) is a function of \(x\), then the derivative of the function \(y\) with respect to \(x\) is \(y' = f(u(x)) \cdot u'(x)\). Substituting the given integral into this formula, we get \(y' = e^{(x^2)^2} \cdot 2x\).

Simplify the Result

Simplify the result to get the final derivative: \(y' = e^{x^4} \cdot 2x\).