Animated Solution for Mathematics - Differentiation: Find the derivative of $\sin (x^2+1)$ with respect to $x$ from first principle.

Welcome, future engineer. Today, we are not just solving a problem; we are peeling back the curtain on how calculus actually works. You have likely used the Chain Rule a thousand times to differentiate $f(x)=\sin (x^2+1)$, but today, we are going to the source using the First Principle.

Imagine you are standing on a curve defined by $f(x)=\sin (x^2+1)$. The derivative is the slope of the tangent line at any point $x$. We find this by taking a secant line connecting point $P$ at $x$ and a nearby point $Q$ at $x+h$.

As $h$ approaches zero, point $Q$ slides toward $P$, and the secant line transforms into the tangent. This is the heartbeat of calculus.

We start with the formal definition of the derivative:

Substituting our specific function, we obtain:

Do not panic at the complexity. First, expand the term $(x+h{)}^2$ to get $x^2+2xh+h^2$. Our limit now takes the form:

To proceed, we must convert the difference of two sines into a product. We invoke the trigonometric identity:

Let $C=x^2+2xh+h^2+1$ and $D=x^2+1$. When we calculate the difference $C-D$, the $x^2$ and $1$ terms vanish, leaving $2xh+h^2$. Dividing this by $2$ yields $h(x+\frac{h}{2})$.

When we calculate the sum $C+D$, we get $2x^2+2xh+h^2+2$. Dividing by $2$ yields $x^2+xh+\frac{h^2}{2}+1$.

Substitute these components back into our limit expression:

We now utilize the standard limit ${\lim }_{\theta \to 0}\frac{\sin \theta }{\theta }=1$. Here, our angle is $\theta =h(x+\frac{h}{2})$.

To match the denominator, we multiply and divide by $(x+\frac{h}{2})$. As $h\to 0$, the term $\frac{\sin (h(x+\frac{h}{2}))}{h(x+\frac{h}{2})}$ approaches $1$.

The cosine term simplifies to $\cos (x^2+1)$, and the remaining factor $(x+\frac{h}{2})$ simplifies to $x$. Multiplying the surviving pieces, we get $2\cdot \cos (x^2+1)\cdot x$.

The final result is:

Look at that elegance. It matches the Chain Rule perfectly, proving that the shortcut you use every day is built on a foundation of pure, logical beauty.