Question

How do you find the derivative of ${4^{6x}}$ ?

Answer 1

In solving the question, first assume the given expression to a variable $y$ , i.e., $y = {4^{6x}}$ .Perform differentiation, This will give us $\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}{4^{6x}}$ , now differentiate this equation by using identity $\dfrac{d}{{dx}}{a^x} = {a^x}\log a$ and chain rule on the RHS, then we will get the required result.

Complete step-by-step solution:
Given expression is ${4^{6x}}$ ,
Now assume the given expression as variable $y$ , we will get,
$y = {4^{6x}}$ ,
Now deriving on both sides we get,
$\dfrac{d}{{dx}}y = \dfrac{d}{{dx}}{4^{6x}}$ ,
Differentiate using the chain rule, which states that $\dfrac{d}{{dx}}\left( {f\left( {g\left( x \right)} \right)} \right) = f'\left( {g\left( x \right)} \right)g'\left( x \right)$ where $f\left( x \right) = {4^x}$ and $g\left( x \right) = 6x$ .
Using chain rule, let $u = 6x$ ,
$\Rightarrow \dfrac{d}{{dx}}{4^u}\dfrac{d}{{dx}}6x$ ,
Now differentiating using exponential Rule which states that $\dfrac{d}{{dx}}{a^u} = {a^u}\log a$ where $a = 4$ , we get,
$\Rightarrow {4^u}\log 4\dfrac{d}{{dx}}6x$ ,
Now substituting the value of u we get,
$\Rightarrow {4^{6x}}\log 4\dfrac{d}{{dx}}6x$ ,
Since $6$ is constant with respect to $x$ , the derivative of $6x$ with respect to $x$ is $6\dfrac{d}{{dx}}x$ .
Now expression becomes,
$\Rightarrow {4^{6x}}\log 4\left( {6\dfrac{d}{{dx}}x} \right)$ ,
Now simplifying we get,
$\Rightarrow 6\left( {{4^{6x}}\log 4\left( {\dfrac{d}{{dx}}x} \right)} \right)$ ,
Now we know that derivative of $x$ with respect to $x$ is 1, the expression becomes,
$\Rightarrow 6 \cdot {4^{6x}}\log 4$ ,
So the derivative of the given expression ${4^{6x}}$ is $6 \cdot \log 4 \cdot {4^{6x}}$ .

$\therefore$ The differentiation value of ${4^{6x}}$ is $6 \cdot \log 4 \cdot {4^{6x}}$ .

Note: Differentiation is the method of evaluating a function’s derivative at any time. Some of the fundamental rules for differentiation are given below, and using these rules we can solve differentiation questions easily.
Sum or difference rule: When the function is the sum or difference of two functions, the derivative is the sum or difference of derivative of each function, i.e.,
If $f\left( x \right) = u\left( x \right) \pm v\left( x \right)$ ,
Then $f'\left( x \right) = u'\left( x \right) \pm v'\left( x \right)$ .
Product rule: When the function is the product of two functions, the derivative is the sum or difference of derivative of each function, i.e.,
If $f\left( x \right) = u\left( x \right) \times v\left( x \right)$ ,
then $f'\left( x \right) = u'\left( x \right)v\left( x \right) + u\left( x \right)v'\left( x \right)$
Quotient Rule: If the function is in the form of two functions $\dfrac{{u\left( x \right)}}{{v\left( x \right)}}$ , the derivative of the function can be expressed as,
$f'\left( x \right) = \dfrac{{u'\left( x \right)v\left( x \right) - u\left( x \right)v'\left( x \right)}}{{{{\left( {v\left( x \right)} \right)}^2}}}$ .
Chain Rule:
If $y = f\left( x \right) = g\left( u \right)$ ,
And if $u = h\left( x \right)$ ,
Then $\dfrac{{dy}}{{dx}} = \dfrac{{dy}}{{du}} \times \dfrac{{du}}{{dx}}$ .