find the equation of the tangent to y=2^x log2x^4 when x=4
To find the equation of the tangent to the curve y = 2^x log2x^4 when x = 4, we need to find the slope of the tangent line at that point and then use the point-slope form of a line to write the equation.
Step 1: Find the slope of the tangent line
The slope of the tangent line to a curve at a specific point is equal to the derivative of the function evaluated at that point. So, we need to find the derivative of the function y = 2^x log2x^4.
Let's start by taking the derivative of each term separately using the product rule and chain rule.
Derivative of 2^x:
Using the chain rule, we have d/dx(2^x) = (ln2) * (2^x).
Derivative of log2x^4:
Using the chain rule, we have d/dx(log2x^4) = (1)/(ln2x^4) * (1/x^4) * (4x^3).
Step 2: Evaluate the derivative at x = 4
Now, substitute x = 4 into the derivative expressions we found earlier to find the slope of the tangent line at x = 4.
For the derivative of 2^x, we have (ln2) * (2^4) = 16ln2.
For the derivative of log2x^4, we have (1)/(ln2 * 4) * (1/4^4) * (4 * 4^3) = 4/ln2.
So, the slope of the tangent line at x = 4 is 16ln2 * (4/ln2) = 64.
Step 3: Write the equation of the tangent line
Now that we have the slope of the tangent line, we can use the point-slope form of a line to write the equation.
The equation of a line in point-slope form is given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line, and m is the slope.
We know that x = 4 is the point of tangency. Let's use this point along with the slope we found (m = 64) to write the equation.
Using the point-slope form, we have y - y₁ = m(x - x₁).
Replacing x₁ with 4 and y₁ with 2^4 log2(4^4), we get:
y - (16 log2(16)) = 64(x - 4).
Simplifying further, we have:
y - 16(1) = 64(x - 4),
y - 16 = 64x - 256.
Therefore, the equation of the tangent to the curve y = 2^x log2x^4 when x = 4 is y = 64x - 240.