Q.Find $\frac{dy}{dx}$ in the following: $\sin (x^2 + 5)$

Concept: Chain Rule — differentiate the outer function, then multiply by the derivative of the inner function.

Let $y = \sin(u)$ where $u = x^2 + 5$.

Step 1: $\frac{dy}{du} = \cos(u)$.

Step 2: $\frac{du}{dx} = 2x$.

Step 3: By the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = \cos(u) \cdot 2x$.

Step 4: Substitute back $u = x^2 + 5$ to get $\frac{dy}{dx} = 2x \cos(x^2 + 5)$.

We have $y = \sin(x^2 + 5)$. This is a composite function — a sine function whose input is not just $x$, but another function $x^2 + 5$. Whenever you have a function inside another function, the Chain Rule is the tool.

The Chain Rule says: if $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$. In words: differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function.

Here, the outer function is $\sin(\cdot)$, whose derivative is $\cos(\cdot)$. The inner function is $g(x) = x^2 + 5$, whose derivative is $2x$.

Let’s apply it step by step.

1. Identify the outer and inner functions.

   Outer: $f(u) = \sin u$, where $u = x^2 + 5$.

   Inner: $u = x^2 + 5$.

2. Differentiate the outer function with respect to its input $u$.

   $\frac{d}{du} \sin u = \cos u$.

   So $f'(g(x)) = \cos(x^2 + 5)$.

3. Differentiate the inner function with respect to $x$.

   $\frac{d}{dx}(x^2 + 5) = 2x$.

4. Multiply the two derivatives.

   By the Chain Rule:

Method: Differentiating a Trigonometric Function of a Polynomial (Chain Rule)

Whenever you need $\frac{dy}{dx}$ of a trig function applied to a non-linear expression, the chain rule breaks the work into two easy pieces.

Steps

Step 1: Identify the outer and inner functions

Write $y = f(u)$ where $u$ is everything inside the trig function (the inner expression), and $f$ is the trig function itself (the outer function).

Step 2: Differentiate the outer function with respect to $u$, keeping $u$ unevaluated

Use the standard derivative (e.g. $\frac{d}{du}\sin u = \cos u$) but do NOT substitute the inner expression's own derivative yet — write the result still in terms of $u$.

Step 3: Differentiate the inner function with respect to $x$

This is usually a simple polynomial derivative.

Step 4: Multiply the two results and substitute back

Replace $u$ with its original expression in $x$ to give the final answer purely in terms of $x$.

Common Mistakes

Mistake 1: Forgetting to multiply by the inner derivative

Why it's wrong: writing $\frac{dy}{dx} = \cos(x^2+5)$ and stopping there ignores the chain rule entirely — the derivative of the inner function $x^2+5$ (which is $2x$) must also be multiplied in. Correct approach: always write out both the outer derivative and the inner derivative as separate steps, then multiply them, rather than trying to do the whole thing in one line.

Mistake 2: Differentiating the inside prematurely, inside the outer function's argument

Why it's wrong: some students write $\cos(2x)$ instead of $\cos(x^2+5)$, mistakenly replacing the inner expression with its own derivative before applying the outer function. The argument of $\cos$ must stay as the original inner expression, $x^2+5$ — only after differentiating the outer function do you separately multiply by the inner derivative. Correct approach: keep the inner expression completely unchanged inside the outer function's derivative; the inner derivative only appears as a separate multiplied factor.