hi sorry the first one was actually
If f(x)=5x^3–2/ x^4
find f'(x)____
and for some reason the other two are wrong but thank you anyway
No problem! Let's find the derivative of f(x) = 5x^3 - 2/x^4.
To find the derivative of a function, we can use the power rule for derivatives and the quotient rule if there is a fraction involved.
Using the power rule, the derivative of x^n is nx^(n-1). Applying this rule to the first term, 5x^3, we get:
d/dx (5x^3) = 3 * 5x^(3-1) = 15x^2.
For the second term, -2/x^4, we need to use the quotient rule. The quotient rule states that the derivative of f(x)/g(x) is given by (f'(x) * g(x) - f(x) * g'(x)) / (g(x))^2.
Applying the quotient rule to our second term, we have:
d/dx (-2/x^4) = [(-2)' * x^4 - (-2) * (x^4)'] / (x^4)^2.
The derivative of -2 is 0, and the derivative of x^4 is 4x^3. Substituting these values into the quotient rule equation, we get:
d/dx (-2/x^4) = (0 * x^4 - (-2) * 4x^3) / (x^4)^2
= (8x^3) / (x^8)
= 8x^3 / x^8
= 8 / x^5.
Therefore, the derivative of f(x) = 5x^3 - 2/x^4 is:
f'(x) = 15x^2 + 8/x^5.
I hope this explanation helps! If you have any further questions, feel free to ask.