1)if y=x³+2x²+7x+8 then dy by dx will be-
2) if y= 1/x⁴ then, dy by dx will be:
3) if y = x² sin x, then dy/dx will be-
To find the derivative of a given function, you can use the power rule and product rule of differentiation. Let's solve each problem step by step:
1) Given y = x³ + 2x² + 7x + 8, we can find dy/dx by taking the derivative term by term:
- The derivative of x³ is 3x².
- The derivative of 2x² is 4x.
- The derivative of 7x is 7.
- The derivative of 8 (which is a constant) is 0.
So, dy/dx = 3x² + 4x + 7.
2) Given y = 1/x⁴, we can find dy/dx by applying the power rule and the chain rule as follows:
- Start by bringing down the exponent: dy/dx = -4/x⁵.
- Apply the chain rule by multiplying by the derivative of the inside function, which is 1 (since the derivative of x is 1).
- Combining the steps, we get dy/dx = -4/x⁵.
3) Given y = x² sin x, we can find dy/dx by applying the product rule:
- The derivative of x² is 2x.
- The derivative of sin x is cos x.
- Then, using the product rule, we have dy/dx = 2x cos x + x² sin x.
These are the solutions for the derivatives of the given functions. Remember that the power rule and product rule are fundamental rules in calculus for finding derivatives.
1) do it the same way I did the previous question 1
2) (bottom derivative top - top derivative bottom ) / bottom squared
( x^4 * 0 - 1*4x^3 )/x*8
= - 4 /x^5
3) x^2 cos x - sin x (2x)
you finish