find the equation of the tangent line to y=4^((x^2)-6x+3) at x=4
when x=4
y = 4^(16-24+3) = 4^-5 = 1/1024
so the point is (4,1/1024)
dy/dx = (2x - 6)(ln4)(4^(x^2 - 6x + 3)
when x=4, dy/dx = 2ln4(1/1024) = ln4/512
so y - 1/1024 = (ln4/512)(x-4)
512y - 1/2 = (ln4)x - 4ln4
(ln4)x - 512y = 4ln4 - 1/2
I'll leave it up to you to convert to y = mx+b form if it is needed.
Why did the scarecrow win an award? Because he was outstanding in his field! Now, let's find the equation of this tangent line. To do that, we need to find both the slope and a point on the tangent line.
First, we'll find the slope by taking the derivative of the function y = 4^((x^2)-6x+3). Using the chain rule, the derivative will be:
dy/dx = ln(4) * 4^((x^2)-6x+3) * (2x - 6)
Now, we can substitute x = 4 into the derivative to find the slope at x = 4:
dy/dx = ln(4) * 4^((4^2)-6*4+3) * (2*4 - 6)
We need to find a point on the tangent line, so let's substitute x = 4 into the original function:
y = 4^((4^2)-6*4+3)
Now we have a point, which is (4, [the y-value we just found]).
So, the equation of the tangent line is y - [the y-value we just found] = [slope] * (x - 4).
To find the equation of the tangent line to the curve y = 4^(x^2 - 6x + 3) at x = 4, we need to find the slope of the tangent line and the coordinates of the point where it touches the curve.
Step 1: Find the derivative of y with respect to x.
To find the derivative, we can use the chain rule. Let's differentiate the base and the exponent separately, and then multiply them together.
dy/dx = d/dx[4^(x^2 - 6x + 3)]
= [ln(4) * 4^(x^2 - 6x + 3)] * [2x - 6] (using the chain rule)
= [2ln(4)(x^2 - 6x + 3)](4^(x^2 - 6x + 3))(x - 3)
Step 2: Find the slope of the tangent line at x = 4.
To find the slope of the tangent line, substitute x = 4 into the derivative.
dy/dx(4) = [2ln(4)(4^2 - 6(4) + 3)](4^(4^2 - 6(4) + 3))(4 - 3)
= [2ln(4)(16 - 24 + 3)](4^(16 - 24 + 3))(4 - 3)
= [2ln(4)(-5)](4^(-5))(4 - 3)
= -10ln(4)(1/1024)
Step 3: Find the coordinates of the point where the tangent line touches the curve.
Substitute x = 4 into the original equation y = 4^(x^2 - 6x + 3).
y(4) = 4^(4^2 - 6(4) + 3)
= 4^(16 - 24 + 3)
= 4^(-5)
= 1/1024
At x = 4, the corresponding y-coordinate is 1/1024.
Step 4: Write the equation of the tangent line using the slope and point coordinates.
Using the point-slope form y - y1 = m(x - x1), where m is the slope and (x1, y1) are the coordinates of the point, we can substitute the values we found:
y - (1/1024) = -10ln(4)(1/1024)(x - 4)
Multiplying both sides of the equation by 1024 to eliminate the fraction:
1024y - 1 = -10ln(4)(x - 4)
Finally, rearranging the equation:
1024y = -10ln(4)(x - 4) + 1
Thus, the equation of the tangent line to y = 4^((x^2)-6x+3) at x = 4 is 1024y = -10ln(4)(x - 4) + 1.
To find the equation of the tangent line to the given function at a specific point, we can use the derivative of the function.
Step 1: Find the derivative of the function y = 4^((x^2)-6x+3).
To do this, we can use the chain rule. The derivative of 4^u with respect to u is (ln 4) * 4^u, and the derivative of u = (x^2) - 6x + 3 with respect to x is 2x - 6.
So, the derivative of y with respect to x is:
dy/dx = (ln 4) * 4^((x^2)-6x+3) * (2x - 6)
Step 2: Evaluate the derivative at the given point x = 4.
Substituting x = 4 into the derivative, we have:
dy/dx = (ln 4) * 4^((4^2)-6(4)+3) * (2(4) - 6)
Simplifying this expression, we get:
dy/dx = (ln 4) * 4^(16-24+3) * (8 - 6)
dy/dx = (ln 4) * 4^(-5) * 2
dy/dx = 2/4^5
Step 3: Find the slope of the tangent line.
The slope of the tangent line is equal to the derivative evaluated at the given point. So, in this case, the slope is:
m = dy/dx = 2/4^5
Step 4: Find the y-coordinate of the point on the function at x = 4.
Substitute x = 4 into the original function:
y = 4^((4^2)-6(4)+3)
y = 4^(16-24+3)
y = 4^-5
Step 5: Use the point-slope form to determine the equation of the tangent line.
The point-slope form of a linear equation is:
y - y1 = m(x - x1)
Plugging in the values we obtained:
x1 = 4
y1 = 4^-5
m = 2/4^5
The equation of the tangent line is:
y - 4^-5 = (2/4^5)(x - 4)
Simplifying the equation further, we get the final result:
y - 4^-5 = (1/1024)(x - 4)