To differentiate the function [tex]\( y = \tan(5x - 4) \)[/tex] with respect to [tex]\( x \)[/tex], we can follow these steps:
1. Identify the outer function and the inner function:
- The outer function is [tex]\( \tan(u) \)[/tex] where [tex]\( u = 5x - 4 \)[/tex].
- The inner function is [tex]\( u = 5x - 4 \)[/tex].
2. Differentiate the outer function:
- The derivative of [tex]\( \tan(u) \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( \sec^2(u) \)[/tex]. Thus, [tex]\( \frac{d}{du} [\tan(u)] = \sec^2(u) \)[/tex].
3. Differentiate the inner function:
- The derivative of [tex]\( 5x - 4 \)[/tex] with respect to [tex]\( x \)[/tex] is 5. Thus, [tex]\( \frac{d}{dx} [5x - 4] = 5 \)[/tex].
4. Apply the Chain Rule:
- According to the chain rule, the derivative of [tex]\( y = \tan(5x - 4) \)[/tex] with respect to [tex]\( x \)[/tex] is given by [tex]\( \frac{dy}{dx} = \frac{d}{dx} [\tan(5x - 4)] = \frac{d}{du} [\tan(u)] \cdot \frac{du}{dx} \)[/tex].
Combining these results, we get:
[tex]\[ \frac{dy}{dx} = \sec^2(5x - 4) \cdot 5 \][/tex]
5. Use the identity for [tex]\( \sec^2(u) \)[/tex]:
- Recall that [tex]\( \sec^2(u) = 1 + \tan^2(u) \)[/tex]. So, [tex]\( \sec^2(5x - 4) = 1 + \tan^2(5x - 4) \)[/tex].
Substituting this into our derivative expression, we have:
[tex]\[ \frac{dy}{dx} = 5 \left( 1 + \tan^2(5x - 4) \right) \][/tex]
6. Simplify the expression:
- The expression [tex]\( 1 + \tan^2(5x - 4) \)[/tex] can also be written as [tex]\( \tan^2(5x - 4) + 1 \)[/tex].
Thus, the final result, after combining all steps, is:
[tex]\[ \frac{dy}{dx} = 5 ( \tan^2(5x - 4) + 1 ) \][/tex]
Which can be rewritten as:
[tex]\[ \frac{dy}{dx} = 5\tan^2(5x - 4) + 5 \][/tex]
So, the derivative of [tex]\( y = \tan(5x - 4) \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ \boxed{5\tan(5x - 4)^2 + 5} \][/tex]
