Question

How do you find $f'(x)$ given $f(x) = 2x + 7$ ?

Answer 1

Start by considering $f(x)$ as the function of $x$. Next step is substitution. Substitute the values in place of the terms to make the equation easier to solve. Then we apply the chain rule to solve the derivative. We will solve the derivatives of each term separately.

Complete step by step answer:
First we will start off by directly applying the chain rule.
$f(x) = 2x + 7$
Now we will differentiate the function with respect to x.
We know that the differentiation of a constant is zero.
Hence, $\dfrac{{d(7)}}{{dx}} = 0$
Also, the differentiation of $2x$ is $\dfrac{{d(2x)}}{{dx}} = 2$
Hence, the value of $f'(x)$ will be,

$$= \dfrac{{d(2x)}}{{dx}} + \dfrac{{d(7)}}{{dx}} \\\ = 2 + 0 \\\ = 2 \\\$$

Therefore, the value of $f '(x)$ will be $2$.
Additional Information: A derivative is the rate of change of a function with respect to a variable.
Derivatives are fundamental to the solution of problems in calculus and differential equations. In general, scientists observe changing systems to obtain rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions.

Note: While substituting the terms make sure you are taking into account the degrees and signs of the terms as well. While applying the power rule make sure you have considered the power with their respective signs. Remember that the derivative of $x$ is $1$ and the derivative of constant is $0$. While applying the product rule, keep the first term as it is and differentiate the second term, then differentiate the first term and keep the second term as it is or vice versa.