find the derivative of y=[(4^(7x))(7^(4x))]/[10^(e)]
without using logarithmic differentiation!
1/10^e is a constant that you can leave in that form, as a multiplier of the derivative.
Use the product rule for the rest.
For the derivative of 4^7x, let 7x = u
d/dx 4^(7x) = d/du 4^u * d(7x)/dx
= 4^u * ln4 * 7
= 7 ln4 * 4^(7x)
d/dx 7^(4x) = 4 ln7* 7^(4x)
Put it all together
To find the derivative of the given function without using logarithmic differentiation, we can apply the rules of exponential and power functions.
Let's start by simplifying the expression:
y = [(4^(7x))(7^(4x))]/[10^(e)]
We can rewrite the expression using the properties of exponents:
y = [(4^7x)(7^4x)]/[10^e]
Now, let's use the quotient rule for differentiation. According to the quotient rule, if we have a function f(x) divided by a function g(x), the derivative is given by:
f'(x)g(x) - f(x)g'(x) / [g(x)]^2
In this case, our f(x) is (4^7x)(7^4x) and our g(x) is 10^e.
f'(x)g(x) = (4^7x)(7^4x) * ln(10^e) (Applying the power rule to f(x))
f(x)g'(x) = (4^7x)(7^4x) * e * ln(10) (Applying the power rule to g(x)))
Simplifying further:
f'(x)g(x) - f(x)g'(x) = [(4^7x)(7^4x) * ln(10^e)] - [(4^7x)(7^4x) * e * ln(10)]
Combining like terms:
f'(x)g(x) - f(x)g'(x) = [(4^7x)(7^4x) * ln(10^e - e * ln(10))]
Finally, the derivative of y is given by:
y' = [(4^7x)(7^4x) * ln(10^e - e * ln(10))] / [10^e]^2
Note that this expression cannot be further simplified without knowing the specific value of x and e.