Answer :
To find the derivative of [tex]\( f(x) = 5 e^{-9x^7 + 5x^{10}} \)[/tex] using the chain rule, follow these steps:
1. Identify the outer and inner functions:
- The outer function is [tex]\( g(u) = 5 e^u \)[/tex].
- The inner function is [tex]\( u(x) = -9x^7 + 5x^{10} \)[/tex].
2. Differentiate the outer function [tex]\( g(u) \)[/tex] with respect to [tex]\( u \)[/tex]:
- The derivative of [tex]\( g(u) = 5 e^u \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( g'(u) = 5 e^u \)[/tex].
3. Differentiate the inner function [tex]\( u(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
- The derivative of [tex]\( u(x) = -9x^7 + 5x^{10} \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( u'(x) = -63x^6 + 50x^9 \)[/tex].
4. Apply the chain rule:
- The chain rule states that the derivative of [tex]\( f(x) = g(u(x)) \)[/tex] with respect to [tex]\( x \)[/tex] is given by [tex]\( f'(x) = g'(u(x)) \cdot u'(x) \)[/tex].
- Substitute [tex]\( g'(u) \)[/tex] and [tex]\( u'(x) \)[/tex] into the chain rule:
[tex]\[
f'(x) = 5 e^{-9x^7 + 5x^{10}} \cdot (-63x^6 + 50x^9)
\][/tex]
5. Simplify the expression:
- Multiply the terms to get the final derivative:
[tex]\[
f'(x) = 5(-63x^6 + 50x^9)e^{-9x^7 + 5x^{10}}
\][/tex]
- Rearrange for clarity:
[tex]\[
f'(x) = 5(50x^9 - 63x^6) e^{5x^{10} - 9x^7}
\][/tex]
Therefore, the derivative of [tex]\( f(x) = 5 e^{-9x^7 + 5x^{10}} \)[/tex] is:
[tex]\[
f'(x) = 5(50x^9 - 63x^6) e^{5x^{10} - 9x^7}
\][/tex]
1. Identify the outer and inner functions:
- The outer function is [tex]\( g(u) = 5 e^u \)[/tex].
- The inner function is [tex]\( u(x) = -9x^7 + 5x^{10} \)[/tex].
2. Differentiate the outer function [tex]\( g(u) \)[/tex] with respect to [tex]\( u \)[/tex]:
- The derivative of [tex]\( g(u) = 5 e^u \)[/tex] with respect to [tex]\( u \)[/tex] is [tex]\( g'(u) = 5 e^u \)[/tex].
3. Differentiate the inner function [tex]\( u(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
- The derivative of [tex]\( u(x) = -9x^7 + 5x^{10} \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( u'(x) = -63x^6 + 50x^9 \)[/tex].
4. Apply the chain rule:
- The chain rule states that the derivative of [tex]\( f(x) = g(u(x)) \)[/tex] with respect to [tex]\( x \)[/tex] is given by [tex]\( f'(x) = g'(u(x)) \cdot u'(x) \)[/tex].
- Substitute [tex]\( g'(u) \)[/tex] and [tex]\( u'(x) \)[/tex] into the chain rule:
[tex]\[
f'(x) = 5 e^{-9x^7 + 5x^{10}} \cdot (-63x^6 + 50x^9)
\][/tex]
5. Simplify the expression:
- Multiply the terms to get the final derivative:
[tex]\[
f'(x) = 5(-63x^6 + 50x^9)e^{-9x^7 + 5x^{10}}
\][/tex]
- Rearrange for clarity:
[tex]\[
f'(x) = 5(50x^9 - 63x^6) e^{5x^{10} - 9x^7}
\][/tex]
Therefore, the derivative of [tex]\( f(x) = 5 e^{-9x^7 + 5x^{10}} \)[/tex] is:
[tex]\[
f'(x) = 5(50x^9 - 63x^6) e^{5x^{10} - 9x^7}
\][/tex]
