Find the derivative of:
y= 5^x + 15/x^6
I'm assuming I need to take the natural log of each side, but I get stuck after that part.
I would split it into two problems and add the result.
y = 5^x
ln y = x ln 5
y'/y = ln 5
y' = y ln 5
y' = 5^x ln 5
then
y = 15 x^-6
y' = 15(-6) x^-7
y' = -90/x^7
so finally
y' = 5^x ln 5 - 90/x^7
To find the derivative of the function y = 5^x + 15/x^6, you can use the rules of differentiation.
Step 1: Start by taking the natural logarithm (ln) of both sides of the equation:
ln(y) = ln(5^x + 15/x^6)
Step 2: Apply the logarithmic rule ln(a + b) = ln(a) + ln(b):
ln(y) = ln(5^x) + ln(15/x^6)
Step 3: Apply the logarithmic rule ln(a^b) = b * ln(a):
ln(y) = x * ln(5) + ln(15/x^6)
Step 4: Differentiate both sides of the equation with respect to x:
(d/dx) ln(y) = (d/dx) (x * ln(5) + ln(15/x^6))
Step 5: Apply the chain rule to differentiate the right-hand side:
(d/dx) ln(y) = ln(5) + (d/dx) (ln(15/x^6))
= ln(5) + (d/dx) (ln(15) - 6ln(x))
= ln(5) - 6 * (d/dx) ln(x)
Step 6: Recall that (d/dx) ln(x) = 1/x. Therefore, the previous step can be simplified to:
(d/dx) ln(y) = ln(5) - 6 * (1/x)
Step 7: Now, differentiate y with respect to x using implicit differentiation. Remember that ln(y) represents the natural logarithm of y:
(1/y) * (dy/dx) = ln(5) - 6 * (1/x)
Step 8: Solve for (dy/dx) - the derivative of y with respect to x:
(dy/dx) = y * (ln(5) - 6 * (1/x))
Step 9: Substitute y back using the original equation:
(dy/dx) = (5^x + 15/x^6) * (ln(5) - 6 * (1/x))
And there you have it! The derivative of y = 5^x + 15/x^6 is (5^x + 15/x^6) * (ln(5) - 6/x).