Question

How do you find and simplify $ f\left( x+h \right)-f\left( x \right) $ and $ \dfrac{f\left( x+h \right)-f\left( x \right)}{h} $ given $ f\left( x \right)=7{{x}^{2}}-1 $ ?

Answer 1

Hint: From the question we have been given that $ f\left( x \right)=7{{x}^{2}}-1 $ . We have been asked to find and simplify $ f\left( x+h \right)-f\left( x \right) $ and $ \dfrac{f\left( x+h \right)-f\left( x \right)}{h} $ . Now, to solve this we have to use some certain transformations and substitutions to get the given expression simplified.

Complete step by step answer:
Now considering from the question we have the expression $ f\left( x \right)=7{{x}^{2}}-1 $ .
First of all, let us find out $ f\left( x+h \right)-f\left( x \right) $
To find $ f\left( x+h \right)-f\left( x \right) $ , as we have already discussed above, we have to use some substitutions.
 $ f\left( x \right)=7{{x}^{2}}-1 $
Here, we have to substitute $ x+h $ in the place of $ x $
By substituting the $ x+h $ in the place of $ x $ , we get
 $ f\left( x \right)=7{{x}^{2}}-1 $ and $ f\left( x+h \right)=7{{\left( x+h \right)}^{2}}-1 $
We have a basic formula in algebra, $ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab $
By using the above formula, we can furthermore simplify the equation.
 $ f\left( x+h \right)=7\left( {{x}^{2}}+2xh+{{h}^{2}} \right)-1 $
 $ \Rightarrow f\left( x+h \right)=7{{x}^{2}}+14xh+7{{h}^{2}}-1 $
Now, we have to find $ f\left( x+h \right)-f\left( x \right) $
 $ f\left( x+h \right)-f\left( x \right)=7{{x}^{2}}+14xh+7{{h}^{2}}-1-\left( 7{{x}^{2}}-1 \right) $
On furthermore simplification, we get the below equation,
 $ f\left( x+h \right)-f\left( x \right)=14xh+7{{h}^{2}} $
 $ \Rightarrow f\left( x+h \right)-f\left( x \right)=7h\left( 2x+h \right) $
Now, we have been also asked to find $ \dfrac{f\left( x+h \right)-f\left( x \right)}{h} $
We already got the value of $ f\left( x+h \right)-f\left( x \right) $ , so simply substitute the obtained value in the above asked question to get the solution.
Now, by substituting the obtained value in it, we get the below equation $ \dfrac{f\left( x+h \right)-f\left( x \right)}{h}=\dfrac{7h\left( 2x+h \right)}{h} $
 $ h $ in both the numerator and denominator of the right side of the equation gets cancelled.
After the cancellation of $ h $ in right hand side of the equation, we get
$\Rightarrow$ $ \dfrac{f\left( x+h \right)-f\left( x \right)}{h}=14x+7h $
Hence, the values which have been asked in the given question are found.
 $ f\left( x+h \right)-f\left( x \right)=7h\left( 2x+h \right) $ and $ \dfrac{f\left( x+h \right)-f\left( x \right)}{h}=14x+7h $.


Note:
 We should be well aware of the substitutions that should be made to get the answer to the given question. We should be very careful while doing the calculation especially in this type of problem. We should be well aware of using the given equation in the question to get the solution. By extension of concept we can say that the derivative of a function is given by $ \dfrac{d}{dx}f\left( x \right)=\dfrac{f\left( x+h \right)-f\left( x \right)}{h} $ .