Differentiate each function
a) y = -3x^2 + 5x - 4
b) f(x) = 6/x - 3/(x^2)
c) f(x) = (3x^2 - 4x)(x^3 + 1)
thank you in advance :)
Now wait a minute, you try first.
is it
a) y'= -6x + 5
b) f'(x)= -6/x^2 + 6/x^3
c) f'(x)= 15x^4 + 6x - 16x^3 - 4
To differentiate each function, we will use the power rule, the sum/difference rule, and the product rule.
a) For the function y = -3x^2 + 5x - 4, we can differentiate each term separately.
First term: -3x^2
To differentiate -3x^2, we bring down the exponent and multiply it by the coefficient (-3):
d/dx(-3x^2) = -3 * 2x = -6x
Second term: 5x
To differentiate 5x, the derivative is just the coefficient (5):
d/dx(5x) = 5
Third term: -4
The derivative of a constant is always zero since a constant does not change with respect to x:
d/dx(-4) = 0
Putting it all together, the derivative of y with respect to x is:
dy/dx = -6x + 5
b) For the function f(x) = 6/x - 3/(x^2), we need to differentiate each term using the rules mentioned earlier.
First term: 6/x
Using the power rule, we bring the exponent down and multiply it by the coefficient (6):
d/dx(6/x) = (6 * (-1))/x^2 = -6/x^2
Second term: -3/(x^2)
Using the power rule, we bring the exponent down and multiply it by the coefficient (-3):
d/dx(-3/(x^2)) = -3 * (-2)/x^3 = 6/x^3
Putting it all together, the derivative of f(x) with respect to x is:
f'(x) = -6/x^2 + 6/x^3
c) For the function f(x) = (3x^2 - 4x)(x^3 + 1), we use the product rule to differentiate.
The product rule states that if we have a function u(x) multiplied by v(x), then the derivative is given by:
d/dx(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)
Let's apply the product rule to our function:
u(x) = 3x^2 - 4x
v(x) = x^3 + 1
Now, we need to find the derivatives of u(x) and v(x).
Derivative of u(x):
d/dx(3x^2 - 4x) = 6x - 4
Derivative of v(x):
d/dx(x^3 + 1) = 3x^2
Using the product rule, the derivative of f(x) is:
f'(x) = (6x - 4)(x^3 + 1) + (3x^2)(3x^2 - 4x)
Simplifying this expression will give you the final answer for f'(x).