Question

How do you find the derivative of $\ln \left( \left( 4{{x}^{2}} \right)+9 \right)$ and what are the intervals for which the results are valid?

Answer 1

Assume the given logarithmic function as y and find the value of $\dfrac{dy}{dx}$ using the basic formula of derivative of log to the base e given as: $\dfrac{d\left[ \ln \left( f\left( x \right) \right) \right]}{dx}=\dfrac{1}{f\left( x \right)}\times \dfrac{d\left[ f\left( x \right) \right]}{dx}$. Now, use the fact that ‘the argument of log should always be greater than 0’ and find the interval of x for which the assumed function is defined. This interval of x will be values for which the results will be valid.

Complete step by step solution:
Here, we have been provided with the function $\ln \left( \left( 4{{x}^{2}} \right)+9 \right)$ and we are asked to differentiate it. Also, we are asked to check the domain values for which the results are valid. Here we are going to use the basic formula of the derivative of natural log to get the answer.
Now, let us assume the given function as y, so we have,
$\Rightarrow y=\ln \left( \left( 4{{x}^{2}} \right)+9 \right)$
Differentiating both the sides with respect to x we get,
$\Rightarrow \dfrac{dy}{dx}=\dfrac{d\left[ \ln \left( \left( 4{{x}^{2}} \right)+9 \right) \right]}{dx}$
Here we have the natural log expression that means log to the base e, where e is nearly equal to 2.71. We know that the derivative of natural log function is given by the formula: $\dfrac{d\left[ \ln \left( f\left( x \right) \right) \right]}{dx}=\dfrac{1}{f\left( x \right)}\times \dfrac{d\left[ f\left( x \right) \right]}{dx}$. So, on comparing $\left( 4{{x}^{2}}+9 \right)$ with f (x), we have,
$\begin{aligned} & \Rightarrow \dfrac{dy}{dx}=\dfrac{1}{\left( \left( 4{{x}^{2}} \right)+9 \right)}\times \dfrac{d\left( \left( 4{{x}^{2}} \right)+9 \right)}{dx} \\\ & \Rightarrow \dfrac{dy}{dx}=\dfrac{1}{\left( 4{{x}^{2}}+9 \right)}\times 8x \\\ & \Rightarrow \dfrac{dy}{dx}=\dfrac{8x}{\left( 4{{x}^{2}}+9 \right)} \\\ \end{aligned}$
Hence, the above relation is our answer.
Now, we need to check the interval of x for which this derivative is defined. We know that the logarithmic function is defined only when its argument is greater than 0. Here we have the argument of log equal to $\left( \left( 4{{x}^{2}} \right)+9 \right)$. Clearly, we can see that the value of this quadratic expression will be greater than 0 for any real value of x. So, the derivative is defined for all the real values of x which can be represented in the interval form as $x\in \left( -\infty ,\infty \right)$.

Note: One may note that the function whose derivative we are finding must be defined. You need to check once the domain values for which the function is undefined and reject those values of x otherwise the answer may be considered wrong or incomplete. Remember the derivatives of basic functions like: logarithmic function, exponential function, trigonometric and inverse trigonometric functions.