To find the first derivative of [tex]\( z = \tan(\sin^7(t)) \)[/tex] with respect to [tex]\( t \)[/tex], we will apply the chain rule of differentiation. Here’s a step-by-step approach:
1. Identify the outer function and its argument:
The outer function is [tex]\(\tan(x)\)[/tex], and its argument is [tex]\(x = \sin^7(t)\)[/tex].
2. Differentiate the outer function:
The derivative of [tex]\( \tan(x) \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( \sec^2(x) \)[/tex]. Therefore,
[tex]\[
\frac{d}{dx} \tan(x) = \sec^2(x).
\][/tex]
3. Apply the outer function's derivative to its argument:
So, we get:
[tex]\[
\frac{d}{dx} \tan(\sin^7(t)) = \sec^2(\sin^7(t)).
\][/tex]
4. Identify the inner function:
The inner function is [tex]\( x = \sin^7(t) \)[/tex].
5. Differentiate the inner function:
The inner function [tex]\( \sin^7(t) \)[/tex] is a composite function itself, and can be written as:
[tex]\[
(\sin(t))^7.
\][/tex]
Using the chain rule again on this inner function:
- First, we differentiate power of [tex]\( \sin(t) \)[/tex], which gives us:
[tex]\[
7(\sin(t))^6.
\][/tex]
- Then, we multiply by the derivative of [tex]\( \sin(t) \)[/tex] itself:
[tex]\[
\frac{d}{dt} \sin(t) = \cos(t).
\][/tex]
6. Combine these derivatives:
By combining the results from steps 3 and 5, we get:
[tex]\[
\frac{d}{dt} \tan(\sin^7(t)) = \sec^2(\sin^7(t)) \cdot 7 (\sin(t))^6 \cdot \cos(t).
\][/tex]
7. Simplify the expression using [tex]\( \sec(x) = 1 / \cos(x) \)[/tex]:
We know that [tex]\(\sec(x) = 1 / \cos(x)\)[/tex], so:
[tex]\[
\sec^2(x) = (1 / \cos(x))^2 = \tan^2(x) + 1.
\][/tex]
Thus, the first derivative of [tex]\( z = \tan(\sin^7(t)) \)[/tex] with respect to [tex]\( t \)[/tex] is:
[tex]\[
\frac{dz}{dt} = 7 (\tan(\sin^7(t))^2 + 1) (\sin(t))^6 \cos(t).
\][/tex]
So, the detailed solution opines that:
[tex]\[
\frac{dz}{dt} = 7 (\tan(\sin^7(t))^2 + 1) \sin^6(t) \cos(t).
\][/tex]
