Find the find the function f given that the slope of the...

Find the find the function $f$ given that the slope of the tangent line to the graph of $f$ at any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=\frac{3 x^{2}}{2 \sqrt{x^{3}-1}} ;(1,1)$$

Key Concepts

Differential Equations

This problem involves finding a function based on its derivative, which defines a differential equation. Differential equations require methods to determine the unknown function whose derivatives satisfy the equation, forming a central part of calculus and mathematical modeling.

Integration by Substitution

The technique of integration by substitution is used to simplify the process of finding antiderivatives when the integrand includes a composite function. By choosing an appropriate substitution, the integral becomes easier to evaluate, which is particularly useful in solving differential equations.

Initial Value Problems

An initial value problem provides an additional condition, typically the value of the function at a specific point, which allows for the determination of the constant of integration. This ensures the uniqueness of the solution among the family of potential antiderivatives.

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