Find the derivative of the function y, below. It may be to your advantage to simplify before differentiating. y = 8x(ln(x)+ln(2))-4x+pi
so y = 8x ln(2x) - 4x + π
dy/dx = 8x(1/x) + 8ln(2x) - 4
= 8 + 8ln(2x) =- 4
= 4 + 8ln(2x)
To find the derivative of the function y = 8x(ln(x) + ln(2)) - 4x + π, we can use the sum and power rules of differentiation.
First, let's simplify the function before differentiating.
y = 8x(ln(x) + ln(2)) - 4x + π
y = 8x(ln(2x)) - 4x + π
To find the derivative, we will differentiate each term separately using the rules of differentiation.
1. Differentiating 8x(ln(2x)):
To differentiate this term, we can use the product rule, which states that d/dx of (uv) = u * dv/dx + v * du/dx, where u represents the first term and v represents the second term.
Applying the product rule, we have:
u = 8x
v = ln(2x)
du/dx = 8 (derivative of 8x with respect to x)
To find dv/dx, we will differentiate ln(2x). The derivative of ln(u) is 1/u, so:
dv/dx = 1/(2x) * 2 (derivative of ln(2x) with respect to x)
Now, we can apply the product rule:
d/dx of (8x(ln(2x))) = u * dv/dx + v * du/dx
= (8x) * (1/(2x) * 2) + ln(2x) * 8
= 4 + 8ln(2x)
2. Differentiating -4x:
The derivative of -4x with respect to x is simply -4.
3. Differentiating π:
The derivative of a constant (π in this case) is zero.
Putting it all together, the derivative of y with respect to x is:
dy/dx = 4 + 8ln(2x) - 4 + 0
Simplifying, we get:
dy/dx = 8ln(2x)
To find the derivative of the function y with respect to x, we can apply the rules of differentiation. First, let's simplify the function:
y = 8x(ln(x) + ln(2)) - 4x + π
Using the properties of logarithms, we can combine the two natural logarithms:
y = 8x(ln(x*2)) - 4x + π
Using the property ln(a * b) = ln(a) + ln(b), we have:
y = 8x(ln(2x)) - 4x + π
Now, we can differentiate each term of the function separately.
1) For the first term, 8x(ln(2x)), we use the product rule of differentiation. The product rule states that the derivative of the product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function.
For 8x, the derivative is 8.
For ln(2x), we use the chain rule of differentiation. The chain rule states that the derivative of the composition of two functions is the derivative of the outer function times the derivative of the inner function. The derivative of ln(x) is 1/x.
Applying these rules, we have:
d/dx [8x(ln(2x))] = 8 * ln(2x) + 8 * 1/(2x) * 2
Simplifying further:
= 8ln(2x) + 8/x
2) For the second term, -4x, we differentiate each term separately, considering -4 as a constant:
d/dx [-4x] = -4
3) For the constant term, π, the derivative is always zero since it doesn't depend on x:
d/dx [π] = 0
Putting all the differentiated terms together, the derivative (dy/dx) is:
dy/dx = 8ln(2x) + 8/x - 4
So, the derivative of the function y = 8x(ln(x) + ln(2)) - 4x + π is dy/dx = 8ln(2x) + 8/x - 4.