Differentiate y = 5cos(3x^2) with respect to x.

Step-by-step solution

1. 1

   Identify the composition. The function is a cosine of something that is not simply $x$, so the chain rule is required. Peel it into layers: a constant multiple $5$, an outer function $\cos(u)$, and an inner function

   $u=3x^{2}$

   Differentiating $\cos(3x^{2})$ as if it were $\cos x$ — giving $-\sin(3x^{2})$ alone — is the error the chain rule exists to prevent.

2. 2

   Pull the constant out. A constant multiple passes straight through differentiation:

   $\frac{dy}{dx}=5\cdot\frac{d}{dx}\left[\cos\!\left(3x^{2}\right)\right]$

3. 3

   Differentiate the outer function, keeping the inside untouched. With $\frac{d}{du}\cos u=-\sin u$, the chain rule gives

   $\frac{d}{dx}\cos u=-\sin(u)\cdot\frac{du}{dx}$

   The argument of $\sin$ stays exactly $3x^{2}$ — the inside is never differentiated inside the sine.

4. 4

   Differentiate the inner function and multiply. By the power rule, $\dfrac{du}{dx}=\dfrac{d}{dx}\left(3x^{2}\right)=6x$, so

   $\frac{dy}{dx}=5\cdot\left[-\sin\!\left(3x^{2}\right)\right]\cdot6x=-30x\sin\!\left(3x^{2}\right)$

   The three numeric factors $5$, $-1$ and $6$ collapse into the single coefficient $-30$.

5. 5

   Verify numerically. A central difference $\frac{y(x+h)-y(x-h)}{2h}$ with $h=10^{-6}$ matches $-30x\sin(3x^{2})$ at $x=0.4$, $1.1$ and $2.0$ to within $10^{-5}$ ✓. A structural check also passes: at $x=0$ the derivative is $0$, which is right because $y=5\cos(3x^{2})$ has a maximum there.