Find the differential coetficients of $$ \frac{5}{7} x^{7}+\frac{1}{3} x^{6}-x^{5} $$

Question

UpStudy AI · Accepted Answer

To find the differential coefficients (derivatives) of the function

$$
f(x) = \frac{5}{7} x^{7} + \frac{1}{3} x^{6} - x^{5},
$$

we will differentiate each term separately using the power rule, which states that if $$ f(x) = ax^n $$, then $$ f'(x) = n \cdot ax^{n-1} $$.

1. Differentiate $$ \frac{5}{7} x^{7} $$:
   $$
   \frac{d}{dx} \left( \frac{5}{7} x^{7} \right) = \frac{5}{7} \cdot 7 x^{6} = 5 x^{6}.
   $$

2. Differentiate $$ \frac{1}{3} x^{6} $$:
   $$
   \frac{d}{dx} \left( \frac{1}{3} x^{6} \right) = \frac{1}{3} \cdot 6 x^{5} = 2 x^{5}.
   $$

3. Differentiate $$ -x^{5} $$:
   $$
   \frac{d}{dx} \left( -x^{5} \right) = -5 x^{4}.
   $$

Now, combine all the derivatives:

$$
f'(x) = 5 x^{6} + 2 x^{5} - 5 x^{4}.
$$

Thus, the differential coefficient of the function $$ \frac{5}{7} x^{7} + \frac{1}{3} x^{6} - x^{5} $$ is

$$
f'(x) = 5 x^{6} + 2 x^{5} - 5 x^{4}.
$$
The differential coefficient is $$ f'(x) = 5 x^{6} + 2 x^{5} - 5 x^{4} $$.

Find the differential coetficients of \( \frac{5}{7} x^{7}+\frac{1}{3} x^{6}-x^{5} \)