Answer :
To find [tex]\( f'(x) \)[/tex] for the function [tex]\( f(x) = 7x^6 \cos^{-1}(x) \)[/tex], we will apply the product rule for differentiation. The product rule states that if you have a function in the form of [tex]\( u(x) \cdot v(x) \)[/tex], then its derivative is given by [tex]\( u'(x)v(x) + u(x)v'(x) \)[/tex].
For [tex]\( f(x) = 7x^6 \cos^{-1}(x) \)[/tex]:
1. Identify the two parts of the product:
- [tex]\( u(x) = 7x^6 \)[/tex]
- [tex]\( v(x) = \cos^{-1}(x) \)[/tex]
2. Find the derivative of each part:
- The derivative of [tex]\( u(x) = 7x^6 \)[/tex] is [tex]\( u'(x) = 42x^5 \)[/tex] (using the power rule).
- The derivative of [tex]\( v(x) = \cos^{-1}(x) \)[/tex] is [tex]\( v'(x) = -\frac{1}{\sqrt{1-x^2}} \)[/tex] (this is a known derivative).
3. Apply the product rule:
[tex]\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\][/tex]
4. Substitute the derivatives and functions:
[tex]\[
f'(x) = 42x^5 \cdot \cos^{-1}(x) + 7x^6 \cdot \left(-\frac{1}{\sqrt{1-x^2}}\right)
\][/tex]
5. Simplify the expression:
[tex]\[
f'(x) = 42x^5 \cos^{-1}(x) - \frac{7x^6}{\sqrt{1-x^2}}
\][/tex]
This gives us the final result for the derivative:
[tex]\[
f'(x) = 42x^5 \cos^{-1}(x) - \frac{7x^6}{\sqrt{1-x^2}}
\][/tex]
Therefore, the correct choice is:
2) [tex]\( 42x^5 \cos^{-1}(x) - 7x^6 \frac{1}{\sqrt{1-x^2}} \)[/tex]
For [tex]\( f(x) = 7x^6 \cos^{-1}(x) \)[/tex]:
1. Identify the two parts of the product:
- [tex]\( u(x) = 7x^6 \)[/tex]
- [tex]\( v(x) = \cos^{-1}(x) \)[/tex]
2. Find the derivative of each part:
- The derivative of [tex]\( u(x) = 7x^6 \)[/tex] is [tex]\( u'(x) = 42x^5 \)[/tex] (using the power rule).
- The derivative of [tex]\( v(x) = \cos^{-1}(x) \)[/tex] is [tex]\( v'(x) = -\frac{1}{\sqrt{1-x^2}} \)[/tex] (this is a known derivative).
3. Apply the product rule:
[tex]\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\][/tex]
4. Substitute the derivatives and functions:
[tex]\[
f'(x) = 42x^5 \cdot \cos^{-1}(x) + 7x^6 \cdot \left(-\frac{1}{\sqrt{1-x^2}}\right)
\][/tex]
5. Simplify the expression:
[tex]\[
f'(x) = 42x^5 \cos^{-1}(x) - \frac{7x^6}{\sqrt{1-x^2}}
\][/tex]
This gives us the final result for the derivative:
[tex]\[
f'(x) = 42x^5 \cos^{-1}(x) - \frac{7x^6}{\sqrt{1-x^2}}
\][/tex]
Therefore, the correct choice is:
2) [tex]\( 42x^5 \cos^{-1}(x) - 7x^6 \frac{1}{\sqrt{1-x^2}} \)[/tex]
