Q.Differentiate w.r.t. $x$, the following function:

1. Write each term as a power and use the chain rule. $\sqrt{3x+2}=(3x+2)^{1/2}$: derivative $=\frac12(3x+2)^{-1/2}\cdot 3 = \dfrac{3}{2\sqrt{3x+2}}$. $\dfrac{1}{\sqrt{2x^2+4}}=(2x^2+4)^{-1/2}$: derivative $=-\frac12(2x^2+4)^{-3/2}\cdot 4x = -\dfrac{2x}{(2x^2+4)^{3/2}}$. Adding, the derivative is $\dfrac{3}{2\sqrt{3x+2}}-\dfrac{2x}{(2x^2+4)^{3/2}}$.
2. With the NCERT convention $\log x = \ln x$, use $\frac{d}{dx}\log_a u = \frac{1}{u\,\ln a}\cdot u'$. Here $u=\log x$, $u'=\frac1x$:

   $\frac{d}{dx}\log_7(\log x) = \frac{1}{(\log x)\,\ln 7}\cdot\frac{1}{x} = \frac{1}{x\,\ln 7\,\log x}.$

   ✓Final answer

   1. $\dfrac{3}{2\sqrt{3x+2}}-\dfrac{2x}{(2x^2+4)^{3/2}}$;
   2. $\dfrac{1}{x\,\ln 7\,\log x}$ (with $\log x=\ln x$)

(i) $\sqrt{3x+2} + \dfrac{1}{\sqrt{2x^2+4}}$

Each term is an outer power wrapped around an inner expression, so we use $\frac{d}{dx}[g(x)]^n = n[g(x)]^{n-1}g'(x)$.

First term: $(3x+2)^{1/2}$.

Second term: $(2x^2+4)^{-1/2}$.

Sum of the derivatives:

(ii) $\log_7(\log x)$

Use the base-$a$ log rule $\dfrac{d}{dx}\log_a u = \dfrac{1}{u\,\ln a}\cdot\dfrac{du}{dx}$, with the standard NCERT convention that $\log x$ denotes the natural logarithm $\ln x$ (so $\frac{d}{dx}\log x = \frac1x$).

Here the base is $a=7$ and $u=\log x$:

Therefore

Method: Differentiating Radical and Logarithmic Composite Functions

This method applies to differentiating expressions built from roots (fractional powers) and logarithms of a base other than $e$, where an inner function is nested inside an outer power or log.

Steps

Step 1: Rewrite radicals as fractional powers

Convert every $\sqrt{\cdot}$ or $\dfrac{1}{\sqrt{\cdot}}$ into $(\cdot)^{1/2}$ or $(\cdot)^{-1/2}$ so the ordinary power rule can be applied directly.

Step 2: Apply the chain rule with the power rule (for radicals) or the base-change log rule (for logs)

For a radical term $[g(x)]^{n}$:

For a log-of-a-log expression $\log_a(u(x))$, use the base-$a$ rule (which reduces to the natural-log derivative scaled by a constant):

Step 3: Combine the pieces and simplify

Substitute the inner function's own derivative $g'(x)$ or $u'(x)$, multiply through, and simplify any resulting fraction — combine terms over a common denominator only if the question asks for a single expression.

Common Mistakes

Mistake 1: Dropping the negative sign on a reciprocal-power (negative-exponent) term

Why it's wrong: writing $\dfrac{1}{\sqrt{2x^2+4}}=(2x^2+4)^{-1/2}$ and then differentiating without carrying the negative exponent through the power rule gives a positive derivative where a negative one is required. Correct approach: keep the exponent explicitly negative throughout — $n=-\tfrac12$ — so the power rule's $n\,[g(x)]^{n-1}$ naturally produces the correct sign.

Mistake 2: Forgetting the outer $\dfrac{1}{\ln a}$ factor when differentiating $\log_a(u)$

Why it's wrong: differentiating $\log_7(\log x)$ as if it were $\log(\log x)$ (base $e$) and skipping the $\ln 7$ in the denominator gives an answer too large by a factor of $\ln 7$. Correct approach: always convert $\log_a u$ to $\dfrac{1}{u\ln a}\cdot u'$ explicitly, writing the base's natural log in the denominator before simplifying anything else.