Find the rate of growth of the function a(x)=19(2.046^x

d a(x)/dx = 19(2.046^x)(ln 2.046)

To find the rate of growth of the function a(x) = 19(2.046^x), we need to calculate its derivative with respect to x. The derivative represents the rate of change or the rate of growth of a function.

To differentiate the function, we can use the constant multiple rule and the chain rule. Let's go step by step:

Step 1: Apply the constant multiple rule:
The constant multiple rule states that if we have a constant multiplied by a function, the derivative of the product is equal to the constant times the derivative of the function. In this case, our function is 2.046^x, and the constant is 19.

Derivative of a(x) = 19 * Derivative of 2.046^x

Step 2: Apply the chain rule:
The chain rule is used when we have a composition of functions. In this case, our function is in the form a(b^x), where b is a constant.

To apply the chain rule, we need to differentiate the base function (2.046^x) with respect to x and multiply it by the derivative of the exponent (x) with respect to x.

Derivative of a(x) = 19 * Derivative of 2.046^x
= 19 * (Derivative of 2.046^x) * (Derivative of x)

Step 3: Differentiate the base function:
The derivative of a function in the form b^x, where b is a constant, is found by multiplying the function by the natural logarithm of the base.

Derivative of a(x) = 19 * (2.046^x) * ln(2.046) * (Derivative of x)

Step 4: Simplify the derivative:
The derivative of x with respect to x is simply 1, so we can remove that term.

Derivative of a(x) = 19 * (2.046^x) * ln(2.046)

Therefore, the rate of growth of the function a(x) = 19(2.046^x) is given by the derivative:

Rate of growth of a(x) = 19 * (2.046^x) * ln(2.046)