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Use product rule. d(uv)/dx = u dv/dx +v du/dxu=x du/dx=1v=e^x dv/dx=e^x
d(xe^x)/dx = xe^x + e^x.1 = e^x(x+1)

To find the value of w, let x = w and y = -32. Substitute these values into the equation of the curve, C: y = (2x-3)^5 => -32 = (2(w) - 3 )^5. Note: the symbol, =>, means "implies that." F...

Firstly we can use the difference rule to split f'(x) into three components which we can consider separately. Then using the knowledge that the integral of x^n is 1/(n+1)*x^(n+1) we get the expression for...

a) We need to differentiate this equation using the quotient rule (Given that it is a fraction with an x term on both the top and bottom of the fraction). We assign the numerator and denominator as follow...

dx/dt = 3(t-1)²dy/dt = 3 + 16t^-3dy/dx=(dy/dt)(dt/dx) dy/dx = 3 + 16t^-3 / 3(t-1)²
At t=2 dy/dx= (3 + 16/8) / 3 = 5/3 Gradient of the normal = -3/5with t=2 y...