Q.Find $\frac{d^2y}{dx^2}$, if $y = x^3 + \tan x$.

Concept: Second Derivative (Existence) — we differentiate twice in succession.

First derivative:

Second derivative (differentiate term by term):

Recall $\frac{d}{dx}(\sec^2 x) = 2\sec x \cdot \sec x \tan x = 2\sec^2 x \tan x$.

Thus:

The question asks for $\frac{d^2y}{dx^2}$, the second derivative of $y$ with respect to $x$. When a function is given explicitly as $y = f(x)$, the second derivative is just the derivative of the first derivative. There’s no chain rule complication here — each term is standard.

Why this works:

The second derivative measures the rate of change of the slope. For $y = x^3 + \tan x$, both $x^3$ and $\tan x$ are differentiable everywhere (except at points where $\tan x$ blows up, but the formula itself is valid wherever the function is defined). We just differentiate twice, carefully handling the derivative of $\tan x$.

1. First derivative Differentiate term by term:

We know $\frac{d}{dx}(x^3) = 3x^2$ and $\frac{d}{dx}(\tan x) = \sec^2 x$.

So

2. Second derivative Now differentiate $\frac{dy}{dx}$:

The first part is easy: $\frac{d}{dx}(3x^2) = 6x$.

For $\frac{d}{dx}(\sec^2 x)$, recall that $\sec^2 x = (\sec x)^2$. Use the chain rule:

(If you prefer, you can also remember the direct formula: $\frac{d}{dx}(\sec^2 x) = 2\sec^2 x \tan x$.)

3. Combine

Method: Computing a Second Derivative by Differentiating Twice

This method applies whenever a function $y=f(x)$ is given explicitly and you need $\dfrac{d^2y}{dx^2}$.

Steps

Step 1: Find the first derivative $\dfrac{dy}{dx}$

Differentiate $y$ term by term using the standard rules (power rule, standard trig derivatives, etc.), producing a new function of $x$.

Step 2: Differentiate the first derivative again, term by term

Treat $\dfrac{dy}{dx}$ as an ordinary function and differentiate it exactly as you would any function — watch for any term that is itself a composite function (like $\sec^2x=(\sec x)^2$), which needs the chain rule.

Step 3: Combine and simplify

Add the individually differentiated terms together. If a term required the chain rule, double-check the extra factor from the inner function was actually included, not just the outer-function derivative.

Common Mistakes

Mistake 1: Dropping the chain-rule factor when differentiating $\sec^2x$

Why it's wrong: $\sec^2x=(\sec x)^2$ is a composite function, so its derivative is $2\sec x\cdot\frac{d}{dx}(\sec x)=2\sec x\cdot\sec x\tan x=2\sec^2x\tan x$ — students often stop after the outer power rule and write just $2\sec x$ or $2\sec x\tan x$, missing one factor of $\sec x$. Correct approach: always treat $\sec^2x$ as "outer square of $\sec x$" and multiply by the derivative of $\sec x$ (namely $\sec x\tan x$).

Mistake 2: Confusing $\left(\dfrac{dy}{dx}\right)^2$ with $\dfrac{d^2y}{dx^2}$

Why it's wrong: the second derivative is the derivative of the derivative, not the square of the first derivative — these are conceptually and numerically completely different quantities. Correct approach: always compute $\dfrac{d^2y}{dx^2}$ by differentiating $\dfrac{dy}{dx}$ as a fresh function, never by squaring it.