To find the derivative of the function [tex]\( f(x) = \sqrt{7x + 9} \)[/tex], follow these steps:
1. Rewrite the function in exponential form:
[tex]\( \sqrt{7x + 9} \)[/tex] can be written as [tex]\( (7x + 9)^{1/2} \)[/tex].
2. Apply the chain rule:
If you have a composite function [tex]\( g(h(x)) \)[/tex], the derivative is given by [tex]\( g'(h(x)) \cdot h'(x) \)[/tex]. Here, let [tex]\( u = 7x + 9 \)[/tex]. Then our function [tex]\( f(x) \)[/tex] becomes [tex]\( f(x) = u^{1/2} \)[/tex], where [tex]\( u = 7x + 9 \)[/tex].
3. Differentiate with respect to [tex]\( u \)[/tex]:
The derivative of [tex]\( u^{1/2} \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( \frac{d}{du} (u^{1/2}) = \frac{1}{2} u^{-1/2} \)[/tex].
4. Differentiate [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\( u = 7x + 9 \)[/tex]. The derivative of [tex]\( 7x + 9 \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( 7 \)[/tex].
5. Combine the derivatives:
By the chain rule, multiply the derivative of [tex]\( u^{1/2} \)[/tex] with respect to [tex]\( u \)[/tex] by the derivative of [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
f'(x) = \left( \frac{1}{2} (7x + 9)^{-1/2} \right) \cdot 7
\][/tex]
6. Simplify the expression:
[tex]\[
f'(x) = \frac{7}{2} (7x + 9)^{-1/2}
\][/tex]
Recall that [tex]\( (7x + 9)^{-1/2} \)[/tex] is the same as [tex]\( \frac{1}{\sqrt{7x + 9}} \)[/tex]. So, we can rewrite the expression as:
[tex]\[
f'(x) = \frac{7}{2 \sqrt{7x + 9}}
\][/tex]
Therefore, the derivative of the function [tex]\( f(x) = \sqrt{7x + 9} \)[/tex] is:
[tex]\[
f'(x) = \frac{7}{2 \sqrt{7x + 9}}
\][/tex]
