Derivatives Solved Examples | Find the derivative of function y = (x⁴ + 4)(x² - 3)

𝐕𝐢𝐝𝐞𝐨 𝐮𝐬𝐞𝐟𝐮𝐥 𝐟𝐨𝐫 : 𝑸𝒖𝒂𝒏𝒕𝒊𝒕𝒂𝒕𝒊𝒗𝒆 𝑨𝒑𝒕𝒊𝒕𝒖𝒅𝒆 𝑬𝒙𝒂𝒎𝒔 | 𝐓𝐲𝐩𝐞 𝐨𝐟 𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧 : 𝐒𝐨𝐥𝐯𝐞𝐝 𝐄𝐱𝐚𝐦𝐩𝐥𝐞𝐬
In this video learn how to find find the derivatives.
𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧 : Find the derivative of function y = (x⁴ + 4)(x² - 3)

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#differentiation
#calculus

𝐅𝐞𝐰 𝐂𝐨𝐧𝐜𝐞𝐩𝐭𝐬
✔️d(xⁿ)/dx = nxⁿ⁻¹
✔️d(eˣ)/dx = eˣ
✔️d(aˣ)/dx = aˣ logₑa
✔️d(constant)/dx = 0
✔️d(eᵃˣ)/dx = aeᵃˣ
✔️d(log x)/dx = 1 / x
✔️A function in the form f(x, y) = 0 is known as implicit function.
✔️When both the variables x and y are expressed in terms of a parameter, the involved equations are called parametric equations.
✔️The process of finding out derivative by taking logarithm in the first instance is called logarithmic differentiation.

𝐒𝐨𝐦𝐞 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧𝐬 𝐚𝐧𝐝 𝐭𝐡𝐞𝐢𝐫 𝐝𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞𝐬
✔️Derivative of function f(x) is f′(x)
✔️Derivative of function xⁿ is nxⁿ⁻¹
✔️Derivative of function eᵃˣ is aeᵃˣ
✔️Derivative of function log x is 1/x
✔️Derivative of function aˣ is aˣ logₑa
✔️Derivative of function c (a constant) is 0

𝐁𝐚𝐬𝐢𝐜 𝐋𝐚𝐰𝐬 𝐨𝐟 𝐃𝐢𝐟𝐟𝐞𝐫𝐞𝐧𝐭𝐢𝐚𝐭𝐢𝐨𝐧
✔️Derivative of function h(x) = c.f(x) where c is any real constant is d{h(x)/dx = c . d{f(x)}/dx
✔️Derivative of function h(x) = f(x) ± g(x) is d{h(x)}/dx = d{f(x)}/dx ± d{g(x)}/dx
✔️Derivative of function h(x) = f(x) . g(x) is d{h(x)}/dx = f(x) . d{g(x)}/dx + g(x) . d{f(x)/dx
✔️Derivative of function h(x) = f(x) / g(x) is d{h(x)}/d(x) = [g(x) . d{f(x)}/dx - f(x) . d{g(x)}/dx] / [g(x)]²
✔️Derivative of function h(x) = f{g(x)} is d{h(x)}/d(x) = d{f(z)}/dx . dz/dx, where z = g(x)

𝐓𝐫𝐚𝐧𝐬𝐜𝐫𝐢𝐩𝐭
Hello! and Welcome back. We now discuss an example from the topic differentiation. The question is  differentiate with respect to x; and the function is y is equal to (x⁴ + 4) times (x² - 3).  So we write down the given function. y equal to (x⁴ + 4) times (x² - 3).  Now there are two possible ways to obtain the derivative. The first way is... we open up the brackets... since both are algebraic terms. We can combine this into a single term and then  take the derivative. The second way is... we can also apply the multiplication or the product rule. I will go by the first method. Let us multiply these two brackets. So we have y is equal  to... [Maths Computation](now multiplying  by x⁴ times x² is x⁶). [Maths Computation]  (x⁴ times -3 is -3 times x⁴). [Maths Computation] 
(4 times x² is 4x² and 4 times -3 is minus 12).  Now we can easily differentiate with respect to 
x. So... differentiating with respect to x we get  dy/dx is equal to derivative of x⁶ minus 3 times derivative of x⁴ plus 4 times derivative of x² minus derivative of constant term i.e. derivative of 12. Now we apply the formula for derivatives. The derivative of x⁶ is 6 times  x⁵; -3 into derivative of x⁴4 is 4 times x³; plus 4 into derivative of x² is 2x  minus derivative of 12. That's a constant term 
so derivative is zero. So finally we have dy by  dx as 6 times x⁵ minus 12 times x³ four two is 8 
plus 8x. So this is the final value for dy by dx.

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