Differentiate.
g(x) = 5e^x√x
product rule:
g ' (x) = (5e^x)(1/2)x^(-1/2) + x^(1/2) (5e^x)
clean it up any way you know how
Well, g(x) is quite the character. Let's take a closer look at this peculiar function.
The function g(x) = 5e^x√x seems to be a mixture of exponential and radical stuff. It’s like a math cocktail with a hint of adventure!
First, we have the exponential term, e^x. This term represents exponential growth or decay, depending on the sign of x. It's like a magic potion that can make things either balloon in size or shrink down to nothingness.
Then, we have the square root of x, √x. This is like a wild ride on a mathematical roller coaster. It can take you up, up, and away, or plunge you into the depths of negative numbers. Just hold on tight!
And lastly, we have the number 5 multiplying the whole shebang. This is just a simple coefficient, like someone throwing in a few extra sprinkles on top of an already tasty treat. It adds a little extra flavor and pizzazz.
So, when you put it all together, g(x) = 5e^x√x is a mathematical masterpiece full of exponential growth, radical twists, and a sprinkle of extra goodness. It's like a circus performance that will leave you bewildered and entertained all at once!
To differentiate the function g(x) = 5e^x√x, we will use the differentiation rules for exponential functions and product rule. Here is the step-by-step process:
Step 1: Rewrite the function as g(x) = 5 * e^x * x^(1/2).
Step 2: Apply the product rule. The product rule states that the derivative of the product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x).
Let's define f(x) = 5e^x and h(x) = x^(1/2).
Applying the product rule, we have:
g'(x) = f'(x) * h(x) + f(x) * h'(x).
Step 3: Compute f'(x) and h'(x).
- The derivative of f(x) = 5e^x with respect to x is simply f'(x) = 5e^x since the derivative of e^x is e^x (the exponential function remains the same).
- The derivative of h(x) = x^(1/2) with respect to x can be found using the power rule: h'(x) = (1/2) * x^(-1/2).
Step 4: Substitute the computed derivatives into the expression from Step 2.
g'(x) = (5e^x) * (x^(1/2)) + (5e^x) * ((1/2) * x^(-1/2)).
Step 5: Simplify the expression if possible.
g'(x) = 5e^x√x + (5/2) * e^x * x^(-1/2).
Therefore, the derivative of g(x) = 5e^x√x is g'(x) = 5e^x√x + (5/2) * e^x * x^(-1/2).
To differentiate the function g(x) = 5e^x√x, we will use the product rule of differentiation. The product rule states that if we have two functions u(x) and v(x), the derivative of their product uv with respect to x is given by the formula:
(d/dx) [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
Let's break down the function g(x) = 5e^x√x into two parts:
u(x) = 5e^x
v(x) = √x
Now, let's find the derivatives of these two parts:
u'(x) = d/dx (5e^x) = 5e^x (since the derivative of e^x is e^x itself)
v'(x) = d/dx (√x) = 1/(2√x) (using the power rule for differentiation)
Now, let's apply the product rule:
(d/dx) [g(x)] = (d/dx) [u(x)v(x)] = u'(x)v(x) + u(x)v'(x)
= (5e^x)(√x) + (5e^x) (1/(2√x))
Simplifying this expression, we get:
(d/dx) [g(x)] = 5e^x√x + 5e^x/(2√x)
Therefore, the derivative of g(x) = 5e^x√x is 5e^x√x + 5e^x/(2√x).