Get comprehensive solutions to your problems with IDNLearn.com. Join our Q&A platform to receive prompt and accurate responses from knowledgeable professionals in various fields.
Sagot :
Certainly! Let's solve the given problem step-by-step.
Given the function:
[tex]\[ f(x) = \frac{5 x^2 \tan x}{\sec x} \][/tex]
we want to find the derivative [tex]\( f'(x) \)[/tex] and then evaluate it at [tex]\( x = 3 \)[/tex].
1. Simplify the function:
We start by simplifying the function. Recall that [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex], which means:
[tex]\[ f(x) = \frac{5 x^2 \tan x}{\sec x} = 5 x^2 \tan(x) \cdot \cos(x) \][/tex]
Since [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex], we can further simplify:
[tex]\[ f(x) = 5 x^2 \cdot \frac{\sin(x)}{\cos(x)} \cdot \cos(x) = 5 x^2 \sin(x) \][/tex]
2. Find the derivative [tex]\( f'(x) \)[/tex]:
Now, we need to differentiate [tex]\( 5 x^2 \sin(x) \)[/tex] with respect to [tex]\( x \)[/tex].
To find the derivative, we use the product rule for differentiation, which states that if you have a function [tex]\( h(x) = u(x) \cdot v(x) \)[/tex], then the derivative [tex]\( h'(x) \)[/tex] is given by:
[tex]\[ h'(x) = u'(x)v(x) + u(x)v'(x) \][/tex]
In our case, let:
[tex]\[ u(x) = 5 x^2 \][/tex]
[tex]\[ v(x) = \sin(x) \][/tex]
Find [tex]\( u'(x) \)[/tex] and [tex]\( v'(x) \)[/tex]:
[tex]\[ u'(x) = \frac{d}{dx}(5 x^2) = 10 x \][/tex]
[tex]\[ v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x) \][/tex]
Now, apply the product rule:
[tex]\[ f'(x) = u'(x)v(x) + u(x)v'(x) \][/tex]
[tex]\[ f'(x) = (10 x) \sin(x) + (5 x^2) \cos(x) \][/tex]
[tex]\[ f'(x) = 10 x \sin(x) + 5 x^2 \cos(x) \][/tex]
3. Evaluate [tex]\( f'(3) \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the derivative [tex]\( f'(x) \)[/tex]:
[tex]\[ f'(3) = 10 (3) \sin(3) + 5 (3)^2 \cos(3) \][/tex]
[tex]\[ f'(3) = 30 \sin(3) + 45 \cos(3) \][/tex]
Therefore, the simplified function's derivative is:
[tex]\[ f'(x) = 10 x \sin(x) + 5 x^2 \cos(x) \][/tex]
And the value of the derivative at [tex]\( x = 3 \)[/tex] is:
[tex]\[ f'(3) = 30 \sin(3) + 45 \cos(3) \][/tex]
Hence, [tex]\( f'(x) = 10 x \sin(x) + 5 x^2 \cos(x) \)[/tex] and [tex]\( f'(3) = 30 \sin(3) + 45 \cos(3) \)[/tex].
Given the function:
[tex]\[ f(x) = \frac{5 x^2 \tan x}{\sec x} \][/tex]
we want to find the derivative [tex]\( f'(x) \)[/tex] and then evaluate it at [tex]\( x = 3 \)[/tex].
1. Simplify the function:
We start by simplifying the function. Recall that [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex], which means:
[tex]\[ f(x) = \frac{5 x^2 \tan x}{\sec x} = 5 x^2 \tan(x) \cdot \cos(x) \][/tex]
Since [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex], we can further simplify:
[tex]\[ f(x) = 5 x^2 \cdot \frac{\sin(x)}{\cos(x)} \cdot \cos(x) = 5 x^2 \sin(x) \][/tex]
2. Find the derivative [tex]\( f'(x) \)[/tex]:
Now, we need to differentiate [tex]\( 5 x^2 \sin(x) \)[/tex] with respect to [tex]\( x \)[/tex].
To find the derivative, we use the product rule for differentiation, which states that if you have a function [tex]\( h(x) = u(x) \cdot v(x) \)[/tex], then the derivative [tex]\( h'(x) \)[/tex] is given by:
[tex]\[ h'(x) = u'(x)v(x) + u(x)v'(x) \][/tex]
In our case, let:
[tex]\[ u(x) = 5 x^2 \][/tex]
[tex]\[ v(x) = \sin(x) \][/tex]
Find [tex]\( u'(x) \)[/tex] and [tex]\( v'(x) \)[/tex]:
[tex]\[ u'(x) = \frac{d}{dx}(5 x^2) = 10 x \][/tex]
[tex]\[ v'(x) = \frac{d}{dx}(\sin(x)) = \cos(x) \][/tex]
Now, apply the product rule:
[tex]\[ f'(x) = u'(x)v(x) + u(x)v'(x) \][/tex]
[tex]\[ f'(x) = (10 x) \sin(x) + (5 x^2) \cos(x) \][/tex]
[tex]\[ f'(x) = 10 x \sin(x) + 5 x^2 \cos(x) \][/tex]
3. Evaluate [tex]\( f'(3) \)[/tex]:
Substitute [tex]\( x = 3 \)[/tex] into the derivative [tex]\( f'(x) \)[/tex]:
[tex]\[ f'(3) = 10 (3) \sin(3) + 5 (3)^2 \cos(3) \][/tex]
[tex]\[ f'(3) = 30 \sin(3) + 45 \cos(3) \][/tex]
Therefore, the simplified function's derivative is:
[tex]\[ f'(x) = 10 x \sin(x) + 5 x^2 \cos(x) \][/tex]
And the value of the derivative at [tex]\( x = 3 \)[/tex] is:
[tex]\[ f'(3) = 30 \sin(3) + 45 \cos(3) \][/tex]
Hence, [tex]\( f'(x) = 10 x \sin(x) + 5 x^2 \cos(x) \)[/tex] and [tex]\( f'(3) = 30 \sin(3) + 45 \cos(3) \)[/tex].
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.