Let f(x)=(-3x^2+6)^4(9x^2+7)^14, Find f'(x).
f'(x)= (-3^2+6)^4 d/dx(9x^2+7)^14+(9x^2+7) d/dx(-3x^2+6)^4
=(-3x^2+6)x14(9x^2+7)^13 d/dx(9x^2+7)+(9x^2+7)^14x4(-3x^2+6)^3 d/dx(-3x^2+6)
=14(-3x+6)^4(9x^+7)^13 (18x)+ 4(9x^2+7)(-3x^2+6)^3 (-6x)
i got stuck
Start with the product rule, and take it slowly.
(d/dx)[(-3x^2+6)^4(9x^2+7)^14]
=(d/dx)[(-3x^2+6)^4](9x^2+7)^14 + (-3x^2+6)^4 (d/dx)[(9x^2+7)^14]
=4(-3x^2+6)^3 (d/dx)[-3x^2+6](9x^2+7)^14 + (-3x^2+6)^4*14(9x^2+7)^13(d/dx)[9x^2+7)
=4(-3x^2+6)^3 (-6x) + (-3x^2+6)^4*14(9x^2+7)^13 (18x)
Now take out the common factors (-3x^2+6)^3(9x^2+7)^13 to get
=(-3x^2+6)^3(9x^2+7)^13[4(-6x)(9x^2+7) + (-3x^2+6)14(18x)]
=(-3x^2+6)^3(9x^2+7)^13[-216x^3+168x + -756x^3+1512x]
=(-3x^2+6)^3(9x^2+7)^13[-972x^3+1344x]
=(-3x^2+6)^3(9x^2+7)^13(-12x)(81x^2-112)
=324x(x^2-2)^3(9x^2+7)^13(81x^2-112)
hey how u get 324x?
(-3x^2+6)^3(9x^2+7)^13(-12x)(81x^2-112)
=3^3(-12x)(-2+x^2)^3(9x^2+7)^13(81x^2-112)
=27*12x(x^2-2)^3(9x^2+7)^13(81x^2-112)
=324x(x^2-2)^3(9x^2+7)^13(81x^2-112)
i got it thanks
You're welcome! :)
To find the derivative of f(x), we can use the product rule. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product u(x)v(x) with respect to x is given by:
d/dx(u(x)v(x)) = u(x)*v'(x) + v(x)*u'(x)
In this case, let's consider u(x) = (-3x^2+6)^4 and v(x) = (9x^2+7)^14. We need to find the derivatives of u(x) and v(x) separately, and then apply the product rule.
First, let's find the derivative of u(x). To differentiate (-3x^2+6)^4, we can use the chain rule. The chain rule states that if we have a function g(x) raised to a power n, then the derivative of g(x)^n with respect to x is given by:
d/dx(g(x)^n) = n*g(x)^(n-1)*g'(x)
In this case, g(x) = -3x^2+6 and n = 4. Let's find the derivative of g(x).
d/dx(-3x^2+6) = -6x
Now, let's substitute these values into the chain rule formula to find the derivative of u(x).
d/dx((-3x^2+6)^4) = 4*(-3x^2+6)^(4-1)*(-6x) = 4*(-3x^2+6)^3*(-6x)
Next, let's find the derivative of v(x). To differentiate (9x^2+7)^14, we can again use the chain rule. Let g(x) = 9x^2+7 and n = 14.
d/dx(9x^2+7) = 18x
Now, let's substitute these values into the chain rule formula to find the derivative of v(x).
d/dx((9x^2+7)^14) = 14*(9x^2+7)^(14-1)*(18x) = 14*(9x^2+7)^13*(18x)
Finally, let's apply the product rule to find the derivative of f(x).
f'(x) = u(x)*v'(x) + v(x)*u'(x)
= (-3x^2+6)^4 * 14*(9x^2+7)^13*(18x) + (9x^2+7)^14 * 4*(-3x^2+6)^3*(-6x)
= 14*(-3x^2+6)^4*(9x^2+7)^13*(18x) + 4*(9x^2+7)*(9x^2+7)^13*(-3x^2+6)^3*(-6x)
Therefore, the derivative of f(x) is 14*(-3x^2+6)^4*(9x^2+7)^13*(18x) + 4*(9x^2+7)*(9x^2+7)^13*(-3x^2+6)^3*(-6x).