Find the differentiation of the following function
a) y=5x^4(x-7)^1/2
b) y=(2(x)^1/2-(3))/(x)^1/2+1
Use the product rule for the first and the quotient rule for the second.
1.
y = (5x^4)(x - 7)^(1/2)
dy/dx = (5x^4)(1/2)(x-7)^(-1/2) + 20x^4 (x-7)^(1/2)
I found most students find the difficulty not in finding that first derivative line, but rather in simplifying it
I don't know how far you are expected to take this "first line" derivative.
Let me know before I continue.
For me to simplify the answer is the most difficult. Especially when meet fraction or with surd..
look for common factors first,
looking at the 5x^4 and 20x^3 , that would be 5x^3
looking at the (x-7)^(-1/2) and (x-7)^(1/2) , that would be (x-7)^(-1/2) , (take the lower exponent one)
also I would take out the fraction of 1/2
= (1/2)(5x^3)(x-7)^(-1/2) [x + (2/1)(4)(x-7) ]
= (5/2)x^3 (x-7)^(-1/2) [ x + 8x - 56]
= 5x^3 (9x - 56)/(2√(x-7) )
How you get (2/1) in the []?
To find the differentiation of a function, we need to apply the rules of differentiation. Here's how you can find the derivatives of the given functions:
a) y = 5x^4(x-7)^(1/2)
To differentiate this function, we can use the product rule and the chain rule.
The product rule states that if we have two functions u(x) and v(x), then the differentiation of their product (u(x) * v(x)) is given by:
d(uv)/dx = u * dv/dx + v * du/dx.
In our case, let's consider u(x) = 5x^4 and v(x) = (x-7)^(1/2).
The derivative of u(x) is du/dx = 20x^3 (using the power rule).
The derivative of v(x) can be found using the chain rule. Let's denote v(x) as w(u) and differentiate w(u) with respect to u first. Then we multiply it by du/dx.
w(u) = u^(1/2)
dw/du = (1/2)u^(-1/2) (using the power rule for differentiation)
Now, let's substitute u = (x-7) into dw/du.
dw/dx = dw/du * du/dx = (1/2)(x-7)^(-1/2) * du/dx = (1/2)(x-7)^(-1/2) * 20x^3
Finally, applying the product rule formula, we get:
dy/dx = 5x^4 * (1/2)(x-7)^(-1/2) * 20x^3 + (x-7)^(1/2) * 20x^3
Simplifying this expression will give us the final derivative.
b) y = (2(x)^(1/2) - 3)/(x)^(1/2) + 1
To differentiate this function, we can use the quotient rule.
The quotient rule states that if we have two functions u(x) and v(x), then the differentiation of their quotient (u(x) / v(x)) is given by:
d(u/v)/dx = (v * du/dx - u * dv/dx) / v^2.
In our case, let's consider u(x) = 2(x)^(1/2) - 3 and v(x) = (x)^(1/2) + 1.
The derivative of u(x) can be found using the sum/difference rule and the power rule.
du/dx = (d(2(x)^(1/2))/dx) - 0
= (2 * (1/2)x^(-1/2)) - 0
= x^(-1/2)
The derivative of v(x) can be found using the power rule.
dv/dx = (d((x)^(1/2))/dx)
= (1/2)x^(-1/2)
Now, we can apply the quotient rule formula:
dy/dx = ((x)^(1/2) + 1) * (x^(-1/2)) - (2(x)^(1/2) - 3) * (1/2)x^(-1/2)) / ((x)^(1/2) + 1)^2
Simplifying this expression will give us the final derivative.
Remember to simplify the expressions and solve for dy/dx in order to obtain the differentiation of the given functions.