Question

How do you find the derivative of \[f(x)={{7}^{2x}}\]?

Answer 1

Hint: To determine the derivative of exponential functions, we need to apply a logarithm on both sides of the function. Solve the equation to make it easier to differentiate and then differentiate the function implicitly. And then rearranging the terms in the equation gives us the derivative of exponential functions which is required.

Complete step by step answer:
As per the given question, we need to find the derivative of the given function using the required properties. Here, the given function is \[f(x)={{7}^{2x}}\].
Let us assume \[y=f\left( x \right)\] then \[y={{7}^{2x}}\].
Now we apply natural log on both sides to get
\[\Rightarrow \ln y=\ln {{7}^{2x}}\]
We know that, according to logarithmic rules \[\ln {{a}^{x}}=x\ln a\]. On substituting this in the above equation, we get
\[\Rightarrow \ln y=2x\ln 7\] ----equation (1)
We know that the derivative of logarithmic function, \[\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}\].
According to power rule, \[\dfrac{d}{dx}(a{{x}^{n}})=(na){{x}^{n-1}}\].
On substituting a=2 and n=1 in the power rule, we get
\[\Rightarrow \]\[\dfrac{d}{dx}(2x)=2\] ----equation (2)
Now we differentiate the equation (1) implicitly using equation (2) to get
\[\Rightarrow \]\[\dfrac{1}{y}dy=2dx\ln 7\]
If we shift \[dx\] to left hand side and shift y to right hand side of the equation we get
\[\Rightarrow \dfrac{dy}{dx}=2y\ln 7\]
Now on substituting y value in above equation we get
\[\Rightarrow \dfrac{dy}{dx}=2({{7}^{2x}})\ln 7\] \[\left( \because y=f(x)={{7}^{2x}} \right)\]
\[\Rightarrow \]\[\dfrac{d}{dx}(f(x))=2({{7}^{2x}})\ln 7\]
\[\therefore \] the derivative of the function \[f(x)={{7}^{2x}}\] is \[2({{7}^{2x}})\ln 7\].

Note:
We can solve this problem easily by remembering the formula \[\dfrac{d}{dx}({{a}^{x}})=({{a}^{x}})\ln a\]. By substituting the values given in question in the formula we can get the solution. It is a single-step process that makes the problem easier to solve. We must remember to find the inner derivative for the functions present in the equation. In order to solve these types of problems, we must have enough knowledge about the derivatives of basic functions. We should avoid calculation mistakes to get the correct solution.