Question

Differentiate with Respect to f

$\huge{ \boxed{\bold \red{ \frac{d}{df} (f {}^{96} + x {}^{5} )}}}$
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Answer 1

Answer:

Answer in The Attechment

Answer 2

${\bold \red{ \frac{d}{df} (f {}^{96} + x {}^{5} )}}$

1. Apply the sum rule

$\small \bold \red{ \frac{d}{df}(f(f) + g(f)) = \frac{d}{df} f(f) + \frac{d}{df} g(f)}$

$\bold \red{f(f) = f {}^{96} }$

$\bold \red{g(f) = {x}^{5} }$

This gives:

${ \boxed{ \bold \red{ \frac{d}{df} ( {f}^{96} ) + \frac{d}{df}( {x}^{5} )}}}$

2. Apply the constant rule to the term $\bold{ \frac{d}{df} }$

Recall the defination of the constant rule

$\bold \red{ \frac{d}{df} C = 0}$

This means:

$\bold \red{\frac{d}{df} ( {x}^{5} ) = 0}$

We can now rewrite the derivative as:

${ \boxed{\bold \red{ \frac{d}{df} (f {}^{96} )}}}$

3. Apply the power rule to the term $\bold{ \frac{d}{df} (f {}^{96} )}$

Recall the defination of the power rule

$\bold \red{ \frac{d}{df} (f {}^{n} ) = nf {}^{n - 1} }$

This means:

$\bold \red{\frac{d}{df} (f {}^{96}) = 96 \times f {}^{95}}$

We can rewrite the derivative as:

${ \boxed{ \bold \red{96 \: f{}^{95}}}}$