differentiate y=3^xlnx
To differentiate the function y = 3^x * ln(x), we can use both the product rule and the chain rule. Here's a step-by-step explanation:
Step 1: Apply the product rule:
The product rule states that if we have a function of the form u(x) * v(x), then the derivative is given by: (u(x)*v'(x)) + (u'(x)*v(x)).
In this case, let's define u(x) = 3^x and v(x) = ln(x).
Step 2: Find the derivatives of u(x) and v(x):
To find the derivative of u(x) = 3^x, we can use the chain rule. The chain rule states that if we have a composite function f(g(x)), then its derivative is given by: f'(g(x)) * g'(x).
In this case, u(x) = 3^x, and the derivative of 3^x with respect to x is: d/dx (3^x) = ln(3) * 3^x.
To find the derivative of v(x) = ln(x), we can use the standard derivative rule for ln(x), which states: d/dx (ln(x)) = 1/x.
Step 3: Apply the product rule:
Now that we have the derivatives of u(x) and v(x), we can use the product rule to differentiate y = u(x) * v(x):
y' = (u(x) * v'(x)) + (u'(x) * v(x))
= (3^x * 1/x) + (ln(3) * 3^x * ln(x))
Therefore, the derivative of y = 3^x * ln(x) is:
dy/dx = (3^x * 1/x) + (ln(3) * 3^x * ln(x))
That's the differentiation of the given function y = 3^x * ln(x).