Q.Differentiate w.r.t. $x$: $\sin^n(ax^2 + bx + c)$.

Concept: Chain Rule — differentiate the outer function (power of sine), then the inner sine, then the quadratic.

Step 1: Let $u = ax^2 + bx + c$, so the function is $(\sin u)^n$.

Step 2: Differentiate using the chain rule:

Step 3: $\frac{du}{dx} = 2ax + b$. Substitute back $u = ax^2 + bx + c$: …

When you see a function like $\sin^n(ax^2 + bx + c)$, the key is to recognise the nesting. You have an outer power function (raising something to the $n$th power), a middle sine function, and an innermost quadratic polynomial. The chain rule says: differentiate from the outside in, multiplying each derivative along the way.

Let’s unpack it step by step.

1. Identify the outermost layer.

   The expression is $[\sin(ax^2 + bx + c)]^n$. The outermost operation is “raise to the power $n$”. So treat the whole inside as a single variable $u = \sin(ax^2 + bx + c)$. Then the derivative of $u^n$ with respect to $u$ is $n u^{n-1}$.

2. Multiply by the derivative of the middle layer.

   Now $u = \sin(v)$, where $v = ax^2 + bx + c$. The derivative of $\sin(v)$ with respect to $v$ is $\cos(v)$. So we multiply by $\cos(v)$.

3. Multiply by the derivative of the innermost layer.

   Finally, $v = ax^2 + bx + c$. Its derivative with respect to $x$ is $2ax + b$.

4. Put it all together.

   Start from the outside: …

Method: Chain Rule for Three-Layer Compositions — Power of a Trig Function of a Polynomial

Use this method whenever a function has the structure: a power, of a trig function, of a polynomial — such as $\sin^n(ax^2+bx+c)$ — three distinct layers requiring three chain-rule multiplications.

Steps

Step 1: Identify the three layers from outside in

Outermost: raising something to the power $n$. Middle: the sine function. Innermost: the polynomial $ax^2+bx+c$. Naming the layers explicitly before differentiating prevents skipping one.

Step 2: Differentiate the outer power layer

Treat $\sin(ax^2+bx+c)$ as a single block $u$, and differentiate $u^n$ using the power rule for functions:

Step 3: Differentiate the middle sine layer …

Common Mistakes

Mistake 1: Dropping the leading factor $n$

Why it's wrong: the power rule for functions requires multiplying by the original exponent $n$ before reducing it to $n-1$; omitting this factor (writing only $\sin^{n-1}(\ldots)\cos(\ldots)(2ax+b)$) leaves the answer missing a constant multiple, which is a full mark loss on a board exam. Correct approach: always write the exponent-rule factor $n$ explicitly as the very first step of differentiating the outer power.

Mistake 2: Applying the exponent $n-1$ to the polynomial instead of to the sine function …